A Demanufacturer's Optimal Acquisition, Pricing, and Inspection Planning Problems under the Trade-Off between Environmen...

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Title:
A Demanufacturer's Optimal Acquisition, Pricing, and Inspection Planning Problems under the Trade-Off between Environmental Fees and Operational Costs
Physical Description:
1 online resource (127 p.)
Language:
english
Creator:
Yildirim, Gonca
Publisher:
University of Florida
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Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
Akcali, Elif
Committee Members:
Geunes, Joseph P
Tufekci, Suleyman
Davis, Timothy A

Subjects

Subjects / Keywords:
demanufacturing -- newsvendor -- yield
Industrial and Systems Engineering -- Dissertations, Academic -- UF
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Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract:
In this dissertation, we investigate the acquisition and allocation decisions, and the value of the yield information for a demanufacturer in his efforts to reclaim environmental and economical values residing in an end-of-life product due to a reusable part on it. Motivated by the current practices and evolving regulatory needs of the demanufacturing industry, we consider the demanufacturer's trade-off between the environmental fees incurred for storing environmentally hazardous end-of-life products unprocessed, and the costs incurred for demanufacturing operations. This dissertation includes three main chapters that contribute to the existing remanufacturing literature.  In the first part, using a three-point-stock setting where acquired end-of-life products can be kept in separate stock points for end-of-life products, disassembled parts, and recovered parts, we formulate a single-period,  profit-maximizing newsvendor problem. We consider acquisition price-dependent supply of end-of-life products, stochastic demand for reusable parts, and stochastic demanufacturing yield rate, and their special cases. We prove the concavity of the expected profits, and using Karush-Kuhn-Tucker (KKT) conditions, characterize the optimal acquisition and stocking policy.  In the second part, we focus on assessing the value of the demanufacturing yield information by comparing two diagnostic testing schemes, and a benchmark case with no testing. The diagnostic test is conducted to eliminate unrecoverable items from being stocked, and may be subject to classification errors. In preliminary testing, all acquired end-of-life products are tested upon acquisition, and before allocation. In postponed testing, the test is conducted on the extracted parts after the disassembly. Under a profit maximizing newsvendor formulation, we use KKT conditions for solution, and present numerical comparisons of the models.  In the final part, we present a two-period extension of our first problem considering the market life cycle of the end-of-life product, and part  commonality between the two consecutive generations of the end-of-life product. We present three myopic policies under deterministic demanufacturing yield, considering preliminary diagnostic testing with perfect accuracy, and two stock points, where disassembly and recovery are combined into a single operation. The optimal solution in each period is a base-stock policy, and depends on the trade-off between the environmental fees and the operational costs.
General Note:
In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Gonca Yildirim.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Akcali, Elif.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-08-31

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ADEMANUFACTURER'SOPTIMALACQUISITION,PRICING,ANDINSPECTION PLANNINGPROBLEMSUNDERTHETRADE-OFFBETWEENENVIRONMENTAL FEESANDOPERATIONALCOSTS By GONCAYILDIRIM ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2012

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c 2012GoncaYldrm 2

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Tomywonderfulfamily,Nihal,H useyin,Bora,Hilal,andDolunayYldrm, fortheirnever-endingloveandsupport, andtomypersonalsunRaduAndreescu, forbringinglove,happinessandawesomemelodiesintomylife 3

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ACKNOWLEDGMENTS IwouldliketorstthankmyadvisorDr.ElifAkcalforherinvaluablepatience, assistance,andcritiqueinconductingmyresearchandwritingthisdissertation.I wouldliketoacknowledgemycommitteemembers,Dr.S uleymanT ufekci,Dr.Joseph P.Geunes,andDr.TimothyAldenDavisforparticipatinginmycommitteeandtheir feedbackthathelpedmeenrichmyresearchproblems.IwouldalsoliketothankDr.Z. PelinBayndrforhercontributiontotheresearchidea,andDr.ColeSmithforbeingin myqualifyingexamcommittee. Iwouldalsoliketothankmyfriends, IbrahimandG unseliKarakayal,SemraA gral TasknandCanerTaskn,fortheirinvaluablefriendshipsandbeinglikeafamilytome. Theyhavealwayskeptcheckinguponme,motivatingandsupportingme.Iwould alsoliketoacknowledgeDincerKonur,GaiaSerraino,SenemCavuso gluCavdar,and AyseMutluDeryaforsupportingmewiththeirinvaluablefriendships.Ialsowould liketothankKelseyAutin,LeahBlais,SeanMatharoo,JohnQuincyPearce,Meghan Sullivan,AndreaSzabo,TaylorWallsofCoffeeCulturewhonotonlyprovidedmewith alovelyenvironmentwhilewritingthisdissertationbutalsotheirmusicalactivitiesand conversationsthathelpedmetakebreaksandrestinbetweenmystudies. Iamgratefultomywonderfulfamily:myparentsNihalandH useyin,mybrother Boraandsister-in-lawHilal,andmyextendedfamilymembers:myuncleCelalandaunt Firdevs,whohavehavegivenmetheirunconditionallove,supportandencouragement throughoutmylife.Mostrecentmemberofmyfamily,Dolunay,youhavebroughtahuge smiletomyfaceandheartwithyourexistence,givingmehopetogetthisdissertation done.YouarethebestfamilyIcouldeverhavehadandthisdissertationwouldnotbe possiblewithoutyou. Finally,IwouldliketothankRaduAndreescuforcomingintomylifeandbeingmy sunwhenIseemedtohavelostmymotivation.Hisamazingloveandsupport,andjoyful melodieshecreatedmixingsongshaveprovidedmethefueltogoon. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................4 LISTOFTABLES......................................7 LISTOFFIGURES.....................................9 ABSTRACT.........................................11 CHAPTER 1INTRODUCTION...................................13 2LITERATUREREVIEW...............................21 3ADEMANUFACTURER'SACQUISITIONPRICE,QUANTITYANDALLOCATION DECISIONSUNDERTHETRADE-OFFBETWEENENVIRONMENTALFEES ANDOPERATIONALCOSTS............................27 3.1BackgroundandMotivation..........................27 3.2ProblemDenition...............................29 3.3ModelDevelopmentandAnalysis.......................31 3.3.1StochasticYieldRate..........................31 3.3.2DeterministicYieldRate........................37 3.3.3AnalysisoftheKKTEquations....................41 3.4ComputationalExperiments..........................42 4ASSESSINGTHEVALUEOFDIAGNOSTICTESTINGANDYIELDINFORMATION INDEMANUFACTURING..............................50 4.1BackgroundandMotivation..........................50 4.2ProblemDenition...............................52 4.3ModelDevelopmentandAnalysis.......................56 4.3.1StochasticYieldRate..........................57 4.3.2DeterministicYieldRate........................65 4.4ComputationalExperiments..........................72 5MYOPICTWO-PERIODNEWSVENDORPOLICIESFORDEMANUFACTURING UNDERLIFECYCLECONSIDERATIONOFREUSABLEPARTS........81 5.1BackgroundandMotivation..........................81 5.2ProblemDenition...............................83 5.3ModelDevelopmentandAnalysis.......................86 5.3.1SAP:Stand-alonePeriods.......................87 5.3.2SAP-CR:Stand-aloneFirstPeriodwithPartCarry-over.......91 5.3.3SAP-RCR:Stand-aloneFirstPeriodwithPartRecoveryandplease Carry-over................................98 5

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5.4ComputationalExperiments..........................99 6CONCLUSIONANDFUTURERESEARCH....................108 ALOWERBOUNDSONTHEEXPECTEDDEMANUFACTURINGYIELDRATE111 BDERIVATIONSFORSINGLE-STOCK-POINTMODELSUNDERDETERMINISTIC DEMANUFACTURINGYIELDRATE........................112 CPROBABILITIESOFNOTSTOCKINGOUTFORSINGLEANDTHREE-STOCK POINTMODELSUNDERUNIFORMDEMAND..................113 DLOWERBOUNDSONTHEEXPECTEDDEMANUFACTURINGYIELDRATE FORDIAGNOSTICTESTINGANDNOTESTINGMODELS...........114 EKKTSOLUTIONSFORACQUISITION-PRICE-INDEPENDENTSUPPLYMODELS UNDERSTOCHASTICDEMANUFACTURINGYIELDRATE..........116 FTOTALACQUISITIONQUANTITYANDPRICEUNDERTESTINGANDNO TESTINGSCENARIOS...............................120 LISTOFREFERENCES..................................123 BIOGRAPHICALSKETCH................................127 6

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LISTOFTABLES Table page 3-1S3S:KKTcasesunderstochasticdemanufacturingyieldrateandgeneraldemand distribution.......................................35 3-2S3S:KKTcasesunderstochasticdemanufacturingyieldrateanduniform U demand.........................................36 3-3D3S:KKTcasesunderdeterministicdemanufacturingyieldrateandgeneral demanddistribution..................................39 3-4D3S:KKTcasesunderdeterministicdemanufacturingyieldrateanduniform U demand.....................................39 3-5Sensitivityofacquisitionorallocationquantitiestosystemparameters......41 3-6Parametersetforthesingle-periodnewsvendormodels..............44 3-7Rankingsofsingle-stock-pointmodelsunderacquisition-price-independent supply..........................................45 3-8Rankingsofsingle-stock-pointmodelsunderacquisition-price-dependentplease supply..........................................45 3-9Rangesofpercentageprotimprovementofthree-stock-pointmodelsover single-stock-pointmodels, p =50 =30 .....................45 4-1Allocationquantitiesofacquiredend-of-lifeproductsamongstockpoints....56 4-2PTS:KKTcasesunderstochasticdemanufacturingyieldrateandgeneral demanddistribution..................................62 4-3DTS:KKTcasesunderstochasticdemanufacturingyieldrateandgeneral demanddistribution..................................63 4-4NTS:KKTcasesunderstochasticdemanufacturingyieldrateandgeneral demanddistribution..................................63 4-5PTS:KKTcasesunderstochasticdemanufacturingyieldrateanduniform, U demand.......................................64 4-6DTS:KKTcasesunderstochasticdemanufacturingyieldrateanduniform, U demand.......................................64 4-7NTS:KKTcasesunderstochasticdemanufacturingyieldrateanduniform, U demand.......................................64 4-8PTD:KKTcasesunderdeterministicdemanufacturingyieldrateandgeneral demanddistribution..................................70 7

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4-9DTD:KKTcasesunderdeterministicdemanufacturingyieldrateandgeneral demanddistribution..................................70 4-10NTD:KKTcasesunderdeterministicdemanufacturingyieldrateandgeneral demanddistribution..................................70 4-11PTD:KKTcasesunderdeterministicdemanufacturingyieldrateanduniform U demand.....................................71 4-12DTD:KKTcasesunderdeterministicdemanufacturingyieldrateanduniform U demand.....................................71 4-13NTD:KKTcasesunderdeterministicdemanufacturingyieldrateanduniform U demand.....................................71 4-14Parametersetforthetestingscenariosinone-periodnewsvendormodels....72 5-1SAP:KKTcases....................................89 5-2SAP:Optimalacquisitionandstockingpolicy....................90 5-3Parametersetforthemyopictwo-periodnewsvendormodels...........100 A-1Lowerboundsontheexpecteddemanufacturingyieldrate...........111 C-1Probabilitiesofnotstockingout = p )]TJ/F39 11.9552 Tf 10.853 0 Td [(v + r = c i )]TJ/F39 11.9552 Tf 10.853 0 Td [(v )]TJ/F39 11.9552 Tf 10.852 0 Td [(v h X Uniform, U .113 D-1Lowerboundsontheexpecteddemanufacturingyieldratefordiagnostic testingandnotestingmodels............................114 E-1ISPTS:KKTcasesunderstochasticdemanufacturingyieldrateandgeneral demanddistribution..................................117 E-2ISDTS:KKTcasesunderstochasticdemanufacturingyieldrateandgeneral demanddistribution..................................118 E-3ISNTS:KKTcasesunderstochasticdemanufacturingyieldrateandgeneral demanddistribution..................................118 E-4ISPTS:KKTcasesunderstochasticdemanufacturingyieldrateanduniform U demand.....................................118 E-5ISDTS:KKTcasesunderstochasticdemanufacturingyieldrateanduniform U demand.....................................119 E-6ISNTS:KKTcasesunderstochasticdemanufacturingyieldrateanduniform U demand.....................................119 8

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LISTOFFIGURES Figure page 3-1Demanufacturingsystem...............................29 3-2Relativeprotimprovementof S3S oversingle-stock-pointmodels, X U,60, p =50 c d + c r =15% c i =0.1% CV Y =0.0304 h p =0 .............47 3-3Relativeprotimprovementof S3S oversingle-stock-pointmodels, X U,60, p =50 c d + c r =15% c i =0.1% CV Y =0.0304 h p =3% ............48 4-1Quantityoftheacquiredend-of-lifeproductsallocatedandsalvagedafterdiagnostic testingisconducted..................................52 4-2Flowofitemsinthesystemundertestingandnotestingscenarios........53 4-3Comparisonofexpectedtotalprotsunderacquisition-price-dependentsupply, stochasticyieldmodels, a b =%,30% c i = L c d + c r = L ........74 4-4Comparisonofexpectedtotalprotsunderacquisition-price-independentsupply, stochasticyieldmodels, c i = L c d + c r = L ....................75 4-5Comparisonofexpectedtotalprotsunderacquisition-price-dependentsupply, stochasticyieldmodels, a b =%,30% c i = L c d + c r = L DT = DT =1 ........................................76 4-6Comparisonofexpectedtotalprotsunderacquisition-price-independentsupply, stochasticyieldmodels, c i = L c d + c r = L DT = DT =1 ...........77 4-7Comparisonofexpectedtotalprotsunderacquisition-price-dependentsupply, stochasticyieldmodels, a b =%,30% c i PT c i DT = L H c d + c r = L DT = DT =1 ...................................79 4-8Comparisonofexpectedtotalprotsunderacquisition-price-independentsupply, stochasticyieldmodels, a b =%,30% c i PT c i DT = L H c d + c r = L DT = DT =1 ...................................80 5-1Flowofacquiredquantityofend-of-lifeproductsinthesystem..........84 5-2Optimalsolutionof CAP-CR 2 basedoninitialinventorylevelandcriticalthreshold values..........................................97 5-3Expectedquantitycarried-over,totalacquisitionquantityinthesecondperiod, andexpectedtotalprots, y 1 =0.90 c 1 c 2 =,10 c r ,1 c r ,2 =,10 X 1 U ,60 X 2 U ,60 p 1 p 2 =,50 ...................101 5-4Expectedquantitycarried-over,totalacquisitionquantityinthesecondperiod, andexpectedtotalprots, y 1 =0.90 c 1 c 2 =,10 c r ,1 c r ,2 =,10 X 1 N ,3 X 2 U ,60 p 1 p 2 =,50 ...................102 9

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5-5Expectedquantitycarried-over,totalacquisitionquantityinthesecondperiod, andexpectedtotalprots, y 1 =0.90 c 1 c 2 =,5 c r ,1 c r ,2 =,10 X 1 U ,60 X 2 U ,60 p 1 p 2 =,50 ...................103 5-6Expectedquantitycarried-over,totalacquisitionquantityinthesecondperiod, andexpectedtotalprots, y 1 =0.90 c 1 c 2 =,5 c r ,1 c r ,2 =,10 X 1 N ,3 X 2 U ,60 p 1 p 2 =,50 ...................104 5-7Expectedquantitycarried-over,totalacquisitionquantityinthesecondperiod, andexpectedtotalprots, y 1 =0.90 c 1 c 2 =,5 c r ,1 c r ,2 =,5 X 1 U ,60 X 2 U ,60 p 1 p 2 =,50 ...................105 5-8Expectedquantitycarried-over,totalacquisitionquantityinthesecondperiod, andexpectedtotalprots, y 1 =0.90 c 1 c 2 =,5 c r ,1 c r ,2 =,5 X 1 N ,3 X 2 U ,60 p 1 p 2 =,50 ...................106 A-1Lowerboundsontheexpecteddemanufacturingyieldrateformodels: S3S C1,C4, S1S-P D3S C1,C4, D1S-P e d :H,H, c :L, b :L.........111 A-2Lowerboundsontheexpecteddemanufacturingyieldrateformodels: ISS3S C1,C4, ISS1S-P ISD3S C1,C4, ISD1S-P e d :H,H, c :L, b :L.....111 D-1Lowerboundsontheexpecteddemanufacturingyieldrateformodels: PTS and PTD C1,C4, e d :H,H, c :L, b :L, =.85,0.80 .........114 D-2Lowerboundsontheexpecteddemanufacturingyieldrateformodels: ISPTS and ISPTD C1,C4, e d :H,H, c :L, b :L, =.85,0.80 .......115 F-1Comparisonoftotalacquisitionquantitiesunderacquisition-price-dependent supply,stochasticyieldmodels, a b =%,30% c i = L c d + c r = L ....120 F-2Comparisonoftotalacquisitionquantitiesunderacquisition-price-independent supply,stochasticyieldmodels, a b =%,30% c i = L c d + c r = L ....121 F-3Comparisonoftotalacquisitionpricesunderacquisition-price-dependentsupply, stochasticyieldmodels, a b =%,30% c i = L c d + c r = L ........122 10

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy ADEMANUFACTURER'SOPTIMALACQUISITION,PRICING,ANDINSPECTION PLANNINGPROBLEMSUNDERTHETRADE-OFFBETWEENENVIRONMENTAL FEESANDOPERATIONALCOSTS By GoncaYldrm August2012 Chair:ElifAkcal Major:IndustrialandSystemsEngineering Inthisdissertation,weinvestigatetheacquisitionandallocationdecisions,andthe valueoftheyieldinformationforademanufacturerinhiseffortstoreclaimenvironmental andeconomicalvaluesresidinginanend-of-lifeproductduetoareusablepartonit. Motivatedbythecurrentpracticesandevolvingregulatoryneedsofthedemanufacturing industry,weconsiderthedemanufacturer'strade-offbetweentheenvironmentalfees incurredforstoringenvironmentallyhazardousend-of-lifeproductsunprocessed,and thecostsincurredfordemanufacturingoperations.Thisdissertationincludesthreemain chaptersthatcontributetotheexistingremanufacturingliterature. Intherstpart,usingathree-point-stocksettingwhereacquiredend-of-lifeproducts canbekeptinseparatestockpointsforend-of-lifeproducts,disassembledparts,and recoveredparts,weformulateasingle-period,prot-maximizingnewsvendorproblem. Weconsideracquisitionprice-dependentsupplyofend-of-lifeproducts,stochastic demandforreusableparts,andstochasticdemanufacturingyieldrate,andtheirspecial cases.Weprovetheconcavityoftheexpectedprots,andusingKarush-Kuhn-Tucker KKTconditions,characterizetheoptimalacquisitionandstockingpolicy. Inthesecondpart,wefocusonassessingthevalueofthedemanufacturingyield informationbycomparingtwodiagnostictestingschemes,andabenchmarkcasewith notesting.Thediagnostictestisconductedtoeliminateunrecoverableitemsfrombeing 11

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stocked,andmaybesubjecttoclassicationerrors.Inpreliminarytesting,allacquired end-of-lifeproductsaretesteduponacquisition,andbeforeallocation.Inpostponed testing,thetestisconductedontheextractedpartsafterthedisassembly.Underaprot maximizingnewsvendorformulation,weuseKKTconditionsforsolution,andpresent numericalcomparisonsofthemodels. Inthenalpart,wepresentatwo-periodextensionofourrstproblemconsidering themarketlifecycleoftheend-of-lifeproduct,andpartcommonalitybetweenthetwo consecutivegenerationsoftheend-of-lifeproduct.Wepresentthreemyopicpolicies underdeterministicdemanufacturingyield,consideringpreliminarydiagnostictesting withperfectaccuracy,andtwostockpoints,wheredisassemblyandrecoveryare combinedintoasingleoperation.Theoptimalsolutionineachperiodisabase-stock policy,anddependsonthetrade-offbetweentheenvironmentalfeesandtheoperational costs. 12

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CHAPTER1 INTRODUCTION Ecologicallychallengingconsumptionpatternsofconsumersinindustrialized societiescontinuetoincreasesolidwastegenerationratesaswellasthefractionof complexindustrialproductsthatentersolidwastestreams.AccordingtotheUnited StatesUSEnvironmentalProtectionAgencyEPA,morethan251milliontonsof municipalsolidwaste,andapproximately7.6billiontonsofindustrialsolidwasteare generatedeachyear.Inaddition,morethan20,000hazardouswastegenerators produceover40milliontonsofhazardouswasteregulatedbyRCRAtheResource ConservationandRecoveryActeachyear. Toreducetheever-increasingeconomicalandenvironmentalburdenofthesolid waste,governmentsareadoptingpoliciestoincreaserecyclingrates.Asaresult,the scopeoftherecyclingindustry,whichwasprimarilylimitedtotherecyclingofsimple consumableproductssuchasaluminumcans,glasscontainers,andplasticbottles inthepast,hasbeenexpandingtoincludethedemanufacturingofcomplexindustrial productssuchascellularphonesandcomputers.Consequently,demanufacturing industryhasevolvedintoamarketwheretheseproductsarenowbeingpurchasedand soldascommodities. Demanufacturingreclaimseconomicalandenvironmentalvaluesresidingin complexend-of-lifeproductsbydisassemblingtheseend-of-lifeproductsandrecovering reusablepartsorrecyclablecomponents.End-of-lifevehiclesalvaging,cellularphone reconditioning,andcomputerrefurbishingareprominentexamplesofdemanufacturing. Demanufacturingofacomplexindustrialproductcanbeaninvolvedprocessthat consistsofseveraloperations.Theseoperationsmayinclude,butnotbelimitedto, thede-pollutionoftheproduct;partialorfulldisassemblyoftheproduct;extractionof reusableand/orrecyclablepartsintheproduct;aswellascleaning,inspection,and recoveryofthesepartstorestorethemintooperatingconditionforreuse. 13

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Someparticulartypesofend-of-lifeproductsmayhavecomponentsormaterials thatposeathreattotheenvironment.Forinstance,end-of-lifevehiclescontain numerouscomponentswithtoxicmaterialse.g.,sodiumazideinairbagsandmercury inconveniencelightsandanti-lockbrakingsystemmodulesaswellasanumberof environmentallyhazardousliquidse.g.,engineoil,transmissionuid,andgasoline. Therefore,thestorageandhandlingofsuchtypesofend-of-lifeproductsmayhaveto beregulatedtoreduceoreliminatepotentialenvironmentaldamageseeforinstance environmentallaw2011;Regulations2011.Onecommonregulationmechanismis tocharge environmentalfees forthestorageofsuchend-of-lifeproducts.Governmental regulationsastohowtoassessthepotentialdamageandcollectenvironmentalfees exhibitsomevariation.InourresearchofthedemanufacturingindustryintheUS, thetwomostcommonformswehavecomeacrossarebasedeitherintheannual processingcapacityof,orthequantityoftheend-of-lifeproductsstoredin,the demanufacturingfacilityseeChapter4inEnvironmentalProtectionAgencyEPA technicalreportbyAnderson2001.Theformerisassessedasaxedcostthatdoes notdependonthenumberofend-of-lifeproductsprocessedinthefacility.Thelatter, however,isassessedasaunitvariablecostthatisbasedonthevolumeorweightofthe pollutant. AcarefulinvestigationoftheoperationsofthedemanufacturingindustryintheUS revealsthatsomedemanufacturersstoreend-of-lifeproductsonly,somestorereusable partsonly,andsomestorebothend-of-lifeproductsandreusableparts.Forinstance, therearesomeautosalvageyardsthatstoreend-of-lifevehicles,whereasthereare othersthatstorereusablepartssuchastransmissionsorengines.Asthedemand forthereusablepartisgenerallyuncertain,thedemanufacturermaywanttopostpone demanufacturingoperationswhicharegenerallylaborintensive;hence,expensive untildemandoccurs.Therefore,thedemanufacturermayprefertostoreend-of-life productsonly.However,iftheend-of-lifeproductisanenvironmentallyobnoxiousone 14

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thatcouldpotentiallyleadtoenvironmentalpollutione.g.,ifend-of-lifevehiclesare storedinopensurfacelotsandexposedtoweathercontinuously,thenthehazardous liquidsinthevehiclemayleakintotheground,pollutingundergroundwatersources, theenvironmentalfeesincurredtostoretheproductmightbequitehigh.Toavoidhigh environmentalfeesinthiscase,thedemanufacturermayprefertostorereusableparts only.Therefore,intheplanningofdemanufacturingoperations,ademanufacturerhasto considerthetrade-offbetweenenvironmentalfeesandoperationalcosts. InChapter3,wefocusontheoptimalacquisitionandallocationdecisionsofa demanufacturersubjecttothetrade-offbetweentheenvironmentalfeesandoperational costs,usingthreestockpointsforallocationoftheacquiredend-of-lifeproducts: end-of-lifeproducts,disassembledbutnotrecoveredparts,andrecoveredparts. AcloselookatthedemanufacturingindustryinUSalsoshowsthatdemanufacturing yieldrateisgenerallya-prioriunknown.Hence,demanufacturersoftenfaceuncertainty abouttherecoverabilityofthereusablepartsonend-of-lifeproducts.Informationabout theconditionofapartmayhelpademanufactureravoidincurringtheenvironmental feesandtheoperationalcostsincasethepartcanbediagnosedasunrecoverable. Suchinformationmaybeobtainedatafeebyconductingadiagnostictest.However, adiagnostictestmaynotalwaysbeaccurate,andmayyieldafalsepositiveora falsenegativeconclusioni.e.,onethatfalselyidentiesapartasrecoverable,and viceversa.Asaresult,someend-of-lifeproductswithactuallyrecoverableparts onthemmaybesalvaged,whereassomeotherswithactuallyunrecoverableparts maybekeptinstockuntiltheirconditionisrevealed.Thedemanufacturerwillincur unnecessarycostsdueto:payingfortheend-of-lifeproductandthensalvagingitinthe former,andincurringenvironmentalfeesiftheend-of-lifeproductiskeptinstockand operationalcostsinthelatter.Hence,demanufacturingyielduncertaintycoupledwith theaforementionedtrade-offbetweentheenvironmentalfeesandoperationalcostsmay necessitateconductingadiagnostictest. 15

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Theaccuracyandstagingofadiagnostictestmayaffectthedemanufacturer's abilitytoefcientlyrecovertheenvironmentalandeconomicalvaluesresidingin end-of-lifeproducts.Thediagnostictestmaybeconducteduponacquisitionof end-of-lifeproductsandpriortoallocationofitemstostockpointsinordertoeliminate someoftheenvironmentalfeesand/orthedemanufacturingcostsbysalvaging theproductsidentiedwithunrecoverableparts.Theseunnecessarycostscanbe eliminatedfullyifthediagnostictestisperfect.However,ifthediagnostictestisnotvery accurate,thenthedemanufacturermayprefertopostponeituntilthepartisextracted ornottoconductitatallinordertopreventlossduetofalseresults.Inbothcases, costofthetest,uncertaintiesassociatedwiththedemanufacturingyieldrate,andthe demandmayalsoaffectthedemanufacturer'sdecisionrelatedtothediagnostictest. Thedemanufacturermayneedtodecidewhethertoconductthediagnostictest,and itsstageifthetestisgoingtobeperformed,dependingontheaccuracyandcostof thetest,valueofthemeandemanufacturingyieldrate,andothersystemparameters. Consequently,analysesandcomparisonsofdifferentdiagnostictestingschemesare requiredtondouttheirimpactonandvalueindemanufacturing. Chapter4utilizestheanalyticalndingsofourrstresearchproblemwherewe useathree-stock-pointsetting,andfocusesonassessingthevalueoftherandom demanufacturingyieldrateusingtwodifferentdiagnostictestingschemessubject toclassicationerrors,andabenchmarkscenariowithnodiagnostictest.Inthe rstdiagnostictestingscenario,whichwecallasthepreliminarytestingscenario,all acquireditemsaretestedbeforeallocationinordertoeliminateunnecessarycostsof keepingthosewithunrecoverableparts.Inthesecondtestingscenario,whichwecallas thepostponedtestingscenario,thediagnostictestisdelayeduntilafterthedisassembly operationiscompleted. Anotherimportantconsiderationindemanufacturingisrelatedtothelimitedmarket lifecycleofend-of-lifeproducts,andhence,ofthereusablepartsonthem.Frequentand 16

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drasticproductand/orpartdesignchangesmaylimitdemanufacturingandsalesofthe samepartovermultipleperiods.Furthermore,duetomarketcompetition,manufacturers keepdesignchangescondentialuntilthenewgenerationoftheproductisready forsale.Consequently,theremaybeuncertaintiesassociatedwiththepartonthe newproduct:thepartmayormaynotreceiveadesignchange,andtheextentofthe designchangemayrendertheoldgenerationofthepartundesirableordifcultto demanufacture,aswellasmakingtheexistingstockoftheend-of-lifeproductsandparts obsolete.Furthermore,adrasticchangeinthepartdesignmayresultinverylittleorno demandfortheoldgenerationofthepart.Insomeothercases,however,thereusable partontheend-of-lifeproductmaybeunchangedorreceiveminordesignupdates,asa resultofwhich,thedemanufacturermaybeabletousehisexistinginventoryoftheolder generation,andmayacquireadditionalend-of-lifeproductsinthenextperiodifneed be.Thisisusuallythecasewhenthereisapartcommonalitybetweentheoldandnew generationsofanend-of-lifeproductwithpossibleminordesignchanges. Whenthereusableparthasadesignchange,thedemanufacturingoperationsmay needmodications,resultingindifferent,andusuallyhigher,demanufacturingcosts. Inaddition,theend-of-lifeproductsacquiredlaterintheirlife-cyclehaveusuallymore tearandwearduetopassageoftimeorusage,increasingthecostsofdemanufacturing forthenewlyacquiredonesinthelaterperiods.Therefore,thedemanufacturermay havetoincurhighercostsofthedemanufacturingoperationsinthenextperiod,despite possiblycheaperacquisitioncostsofsuchend-of-lifeproducts,ascomparedtousing theexistingstockoftheolderbutdeteriorateditemsduetotearandwear,andincurring theassociatedinventoryholdingcostsforcarryingtheleft-overstockintothenext period.Consequently,thedemanufacturermayhavetoconsiderhowmuchtoacquire andstockforthecurrentperiodaswellaswhattodowithhisstockofoldergeneration oftheproductand/orpartinthefutureperiod.Inthiscontext,existenceandmagnitude oftheenvironmentalfeesmaydiscouragethedemanufacturerfromcarryinghisexisting 17

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stockofend-of-lifeproductsintothenextperiodtomeetdemandinthisperiodsince thesefeeswillbeincurredagain.Therefore,thedemanufacturermaychoosetoeither processtheend-of-lifeproductsremaininginstockattheendoftheperiod,incurring theassociateddemanufacturingcosts,andcarryingonlytherecoveredpartsintothe secondperiodincurringtheinventoryholdingcosts,orselltheend-of-lifeproducts remaininginstockattheendoftheperiodtoamaterialrecyclerfortheirmaterial content,earningthesalvagerevenues. Thedecisionsofhowmuchtoacquireandstock,consideringthecurrentand thefutureperiods,maybequitecomplicatedundertheuncertaintyassociated withthenewgenerationoftheend-of-lifeproductand/orthepartcoupledwiththe uncertaintyinthesupplyanddemandduringbothperiods.InChapter5,weinvestigate ademanufacturer'stwo-periodextensionofourrstresearchproblemwithinventory carry-over,wherewesimplifytheproblemsettingbycombiningthedemanufacturing operationsofdisassemblyandrecovery,andhencetheassociatedstockpoints.We investigatethreemyopicpoliciestomaximizeademanufacturer'sexpectedprotsovera two-periodhorizon,inwhichtherstperiodisconsideredasastand-aloneperiod.Inthe rstpolicy,whichwecallasstand-aloneperiods,weassumethatthesecondperiodis alsoastand-aloneperiod,i.e.,allitemsremainingintheinventoryattheendoftherst periodaresoldtoamaterialrecycler.Inthesecondpolicy,whichwecallasstand-alone rstperiodwithpartcarry-over,thedemanufactureroptimizeshisexpectedprotsin therstperiodassumingnoinventorycarry-overintothesecondperiod,butinreality hecarriestheleft-overinventoryofrecoveredpartsintothesecondperiod,salvaging anyexistingend-of-lifeproductsfromtherstperiod.Thethirdpolicy,whichwecallas stand-alonerstperiodwithpartrecoveryandcarry-over,issimilartothesecondpolicy where,thistime,theend-of-lifeproductsremaininginstockattheendoftherstperiod aredemanufactured,andarecarried-overintothenextperiodalongwiththealready recoveredparts. 18

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Theremainderofthisdissertationisorganizedasfollows.InChapter2,weprovide therelevantliterature.InChapter3,wepresentaprotmaximizingnewsvendor modelinordertodeterminetheoptimalacquisitionandallocationdecisionsofthe demanufacturerunderthetrade-offbetweenenvironmentalfeesandoperational costs.Weshowthattheexpectedtotalprotfunctionisstrictlyconcaveunder parametervaluesthatensureprotabilityofthedemanufacturingbusiness,and proposeasolutionusingKKTconditions.WefurthershowthattheKKTconditionsare sufcientandnecessaryduetoconcavityincharacterizingtheoptimalacquisitionand allocationpolicy.Wefurtherpresentandsolvespecialcases,inwhichweconsider acquisition-price-independentsupplyofend-of-lifeproducts,anddeterministic demanufacturingyieldrate.Forallmodels,wealsoprovidelowerboundsonthe expecteddemanufacturingyieldratenecessaryforaprotabledemanufacturing business. InChapter4,wemodifyourformulationsofthree-stock-pointmodelsinourrst researchproblemtoincludeimperfectdiagnostictestingatdifferentstagesofthe demanufacturingoperations,bywhichweassessthevalueofadiagnostictesting, andhence,thedemanufacturingyieldrateinformation.Weshowthattheexpected totalprotsremainstrictlyconcave,andthattheKKTconditionscanstillbeusedto solvefortheoptimalacquisitionandallocationquantitiesineachperiod.Wecompare, numerically,twoimperfecttestingscenarios,inwhichthediagnostictestisconducted atdifferentstagesofdemanufacturingwitheachother,andalsowithabenchmark scenariowithnodiagnostictesting.Wedemonstratetheimportanceoftheaccuracy, staging,andcostsofthetestingscheme,andtheproblemparameters'effectsonthe expectedprotswithanumericalstudy. InChapter5,wepresentthreemyopicpoliciesextendingthesingle-periodmodel inourrstresearchproblemtotwo-periods,consideringacquisition-price-independent supplyanddeterministicdemanufacturingyieldratetoallowinventorycarry-over,and 19

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toincludelife-cycleconsiderationsoftheend-of-lifeproductsandthereusablepartson them.Weshowthateachmyopicmodelcanbesolvedoptimallyforeachperiodusing theKKTconditions,whichleadstosimpledecisionrulesbasedonthetrade-offbetween theenvironmentalfeesandtheoperationalcosts,andtheproblemparameters.Wealso presentanumericalstudycomparingthemyopicpolicies. Chapter6presentsourconcludingremarksabouttheresearchproblemspresented inthisdissertationandpossiblefutureresearchtopics. Byspecicallytakingintoaccountvepracticalconsiderationsindemanufacturing: ievolvingmarketingpracticesforend-of-lifeproducts,iienvironmentalregulations, iiistockingpracticesofdemanufacturers,ivvalueofthedemanufacturingyield informationanddiagnostictestingsubjecttoclassicationerrors,andvmarket life-cycleconsiderationofend-of-lifeproductsandreusablepartsontheseproducts, thathavenotbeenornotjointlyconsideredintheexistingliterature,wepresent realisticandoperationalmodelsfordemanufacturingthatcontributetotheliterature ofremanufacturingandrecycling. 20

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CHAPTER2 LITERATUREREVIEW Productrecoveryandremanufacturinghavereceivedconsiderableattentionand wereferthereadertoAkcalandCetinkaya2011;DekkerandvanderLaan2003; GuideandVanWassenhove2003;GungorandGupta1999;Inderfurthetal.2004; InderfurthandTeunter2003;vanderLaanetal.2004forseveralcomprehensive reviewsoftheliterature.Inthisdissertation,weanalyze,rst,theacquisitionand allocationdecisions;then,theimportanceofthediagnostictestinganddemanufacturing yieldinformation;andlastlytheacquisitionandallocationdecisionsundermarketlife cycleconsiderationsofend-of-lifeproductsovertwoperiodsforademanufacturer whoissubjecttothetrade-offbetweenenvironmentalfeesandoperationalcosts understochasticdemanufacturingyieldrate,stochasticdemandforreusablepartsand acquisition-price-dependentend-of-lifeproductsupply.Therefore,thepertinentliterature canbereviewedunderfourbroadstreams:legislation/regulationofend-of-lifeproducts; acquisition,stockingand/ordisassembly/recoverystrategies;remanufacturingyield and/orqualityclassication;andlifecycleconsiderationsofend-of-lifeproducts. Thestreamofliteraturethatconcentratesonenvironmentallegislationofend-of-life productsdiscussesdifferentpolicies,interrelationsbetweenthelegislationandthe desirabilityofremanufacturingactivities,productdesignand/oroperationaldecisions. CalcottandWalls2000;FullertonandWu1998;PalmerandWalls1999;Walls 2006amongothersdiscussvariousimplementationsofenvironmental/disposal feecollectionpolicies.Atasuetal.2009; Ozdemiretal.2011provideeconomic modelstoanalyzetheeconomicandenvironmentalefciencyoftheextended producerresponsibilitylegislationandefciencyconditionsassumingpredetermined remanufacturingtargets. Thesecondstreamofworkfocusesonacquisitionandstockingdecisions,and disassembly/recoverystrategies.InderfurthandLangella2006;Krikkeetal.1998; 21

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Langella2007;TalebandGupta1997;Teunter2006,amongothers,takethe productordisassemblystructurewithmultiplepartsormultipleproductswithcommon partsintoaccount,andprovideacquisitionand/ordisassembly/recoverystrategiesor schedules.Inthesestudies,simplicationsarerequiredorheuristicsaredevelopeddue toincreasedcomplexityoftheproblemsandtheirsolutions. Thethirdstreamofresearchthatfocusesonremanufacturingyieldand/orquality classicationanalyzestheireffectsoninventory/remanufacturingplanning,pricing and/orcosts.Yieldratehasbeenextensivelycoveredinliteratureforbothforwardand reverseproductionandsupplychainsystems.Fortheformer,wereferthereaderto reviewsofstochasticyieldmodelsbyKhouja1999;YanoandLee1995.Incase ofreversesupplychains,wereviewhereinonlytheportionofproductrecoveryand remanufacturingliteraturefocusingontheremanufacturingyieldandclassicationof end-of-lifeproductsorreturnsbasedontheirquality/remanufacturability. Theportionofremanufacturingandproductrecoveryliteraturethatanalyzesthe impactsoftheyieldrateandconditionoftheend-of-lifeproductsorreturnsonbusiness performancecanbereviewedundertwosub-categories.Therstlineofworkfocuses ontheoptimizationofsomedecisionvariablestooptimizeeitherthesystemcostor protunderagivenremanufacturingyieldinformationand/orqualitybasedclassication ofend-of-lifeproductsorreturns.Thesecondlineofworkfocusesmoreonthecondition oftheend-of-lifeproductsorreturns,andanalyzesthevalueoftheyieldrateand/or qualityinformationontheremanufacturingbusinessperformance. Intherstlineofwork,Guideetal.2003,BakalandAkcali2006,Karakayali etal.2007,andKarakayaletal.2010studythepricingproblemoftheremanufactured items.Guideetal.2003provideaheuristicbymatchingdemandandreturnsin ordertoobtainoptimalsellingpricesasafunctionoftheacquisitionpriceformultiple qualitytypes.BakalandAkcali2006;Karakayalietal.2007;Karakayaletal.2010 investigatetheeffectsofrecoveryyieldrateonpricingdecisionswheresupplyofthe 22

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usedproducts,demandfortherecoveredpartsarepricesensitive,andyielddepends ontheacquisitionprice.Theyinvestigate,bycomparingthreesingle-periodmodels,the benetsofdelayingpricingdecisionsuntilactualyieldisobserved,andthevalueofthe perfectyieldrateinformation.Althoughadditionalrealisticaspectsofacquisitionand marketingdecisionsareincorporatedintheseworks,environmentalconsiderationsare nottakenintoaccount. Thesecondlineofresearch,whichfocusesontheconditionoftheend-of-life productsorreturns,analyzestheimpactofthisknowledge,itssourceoritstimingon inventoryandremanufacturingplanningactivities,costsand/orprots,andprovides theconditionsunderwhichthisinformationisbenecial.Arasetal.2004consider ahybridsystemwithremanufacturingandmanufacturinginordertoassessthecost effectivenessofincorporatingqualitybasedclassicationintoremanufacturingand disposaldecisions.Theauthorsndthatcostsavingsarethegreatestwhenthequality ofthereturnsdecreasesandreturnratesarehigh;anditisgenerallyabetterstrategy toprioritizehigherqualityreturnstomeetdemandthenthelowqualityones.Ina similarsetting,BehretandKorugan2009provideasimulationmodeltoassessthe costsavingsunderseparaterecoveryyieldratesformultiplequalitytypes.Dobosand Richter2006suggestthatconductingqualitybasedinspectionatthefacilityratherthan outsourcingitismoreeconomicalunderaninventoryholdingcostminimizingeconomic orderquantityEOQmodel.Whentotalcostsareconsidered,theauthorsndthat theoppositeistrue.GalbrethandBlackburn2006deriveoptimalacquisitionand sortingpoliciesi.e.,howselectivetobeduringsortinginasingle-periodsettingunder deterministicandstochasticdemandcases.Theyassumethattheremanufacturing costdependsontheconditionoftheusedproducts,andtheconditionisassumedtobe continuousonascale.Theyderivethemaximumremanufacturingcostleveltosatisfy arequiredyieldandanacquisitionquantity.GalbrethandBlackburn2010providean extensionforthisstudy,wherereturnscanhaveuncertainandmultipleconditions. 23

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Ferrer2003focusesonthelotsizingdecisionforpartrecoveryunderrandom yieldratebutdeterministicdemand,allowingoutsidepurchasingoptionwithaleadtime. Theydevelopoptimalpoliciesforfourscenariosaboutthenatureandavailabilityofthe yieldinformation,andprovideaheuristicformulti-partproducts.Theyshowthathaving earlyyieldinformationreducescostssignicantly.FerrerandKetzenberg2004extend thisresearchtomultiple-partproductsininnitehorizon.Ketzenbergetal.2004also provideassessmentofthevalueoftheyieldinformationwheretheyieldismodeledas aBernoullirandomvariable.TeunterandFlapper2011analyzeone-perioddecisions foracquisitionandremanufacturingquantitiesundermultiplequalitytypesbasedon multinomialdistribution,wheretheyieldinformationisavailableuponsorting.They provideaheuristicforthestochasticdemandcase.Fergusonetal.2009useagreedy heuristictosolveastochasticdynamicprograminordertocompareprotgainsfrom qualitygradingovernograding,underbothcapacitatedanduncapacitatedsettings wherenewandremanufacturedproductsarenotsubstitutes.Noneofthelistedworks listedinthisstreamofliteraturesofarconsidersenvironmentalfeesoracquisition price-dependentsupplyofend-of-lifeproducts. ZikopoulosandTagaras2007,2008providesingle-periodprot-maximizing newsvendormodelsunderinnitesupplyofreturnsinordertoderiveprocurementand productionquantities.Althoughtherstpaperismorerelevantfortherstsub-category, wediscussithereasthesecondworkhasthesamesetting.Inthispaper,theauthors consideraserialproductionsystemwithtworeturncollectionsiteswithajointyield distributionfunctionfromwhichtheydeterminethequantitiesofitemstotransport toarefurbishingsite.Alltransporteditemsaresortedattherefurbishingsite.Those failaredisposedwhereasaquantitytorefurbishfromtheresthastobedecidedonto meetastochasticdemand.Inadditiontotransportationandrefurbishingquantities, authorsalsoprovideconditionsunderwhichhavingasingleortwocollectionsitesis optimal.Intheirlatterpaper,theauthorsusethesamesingle-periodnewsvendorsetting 24

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butwithonecollectionsiteanddifferentdisposalcostsfortheremanufacturableand non-remanufacturableitemstofocusmoreonthesortingschemeitself.Theycompare, economically,sortingpriortoremanufacturingversusnosorting,wheresortingissubject tomisclassicationerrors.Theyshowthatsortingmaybeattractivedependingon thesortingaccuracyandtheexpectedyieldofthereturnedproductsanddoesnot dependontheyieldvariability.Theyalsoprovideamulti-periodgeneralizationusinga continuousreviewsystemwithareorderpoint. Thelaststreamofliteraturethatconcentratesonthelifecycleofend-of-lifeproducts mainlyfocusesonthelimitedproductdurabilitysee,forinstance,AtasuandCetinkaya 2006,orBassdiffusionmodelseeBass1969;Deboetal.2006;Geyeretal. 2007.Inourwork,wefocusonademanufacturerwhooperatesindependentlyof themanufacturer,andacquiresend-of-lifeproductsfromtheend-usermarket.We investigateatwo-periodextensionofthesingle-perioddemanufacturingoperations, whereeachperiodmodelisbasedonanewsvendortypeofformulation.Foran extensivecoverageofthenewsvendormodelsinsingleandmulti-periodsettings, wereferthereadertoVeinott1965a,Veinott1965b,andVeinottandWagner1965. Ourcontributiontothepertinentreversesupplychainliteratureisincorporation ofrealisticaspectsfromthecurrentanddevelopingpracticesinthedemanufacturing industryintooperationalmathematicalmodelsalongwithenvironmentalconsiderations thathavenotbeenjointlyconsideredintheexistingliterature.Wespecicallytake intoaccountthetrade-offbetweentheenvironmentalfeesandoperationalcosts,and theacquisitionprice-dependentsupplyofend-of-lifeproductswhileincorporating uncertaintiesthatpertaintodemandforreusableparts,andthedemanufacturingyield rate.Usinganewsvendorsetting,weprovidemathematicalmodelsthatincorporate theseaspectsinordertomaximizeademanufacturer'sexpectedprots.Weshowthat KKTconditionscharacterizetheoptimalacquisitionandallocationpolicy,andcanbe usedtocompare,numerically,theimpactofdifferentdiagnostictestingschemesonthe 25

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systemprots.Furthermore,theseconditionsleadtosimpledecisionrulesthatcanbe solvedveryefcientlyineachperiodundermyopicpoliciesforatwo-periodextension ofthenewsvendorformulation,whenthesupplyquantityofend-of-lifeproductsis independentoftheacquisitionpriceandthedemanufacturingyieldrateisdeterministic. Althoughweconsideranewsvendorsetting,similartosomeoftheexistingwork thatconsiderremanufacturingyieldorqualityclassication,thesettingweuseisnot aserialproductionsystem,andourresearchfocusesonamoreoperationalsetting withadditionalaspects.Specically,weassumethattheend-of-lifeproductsacquired isallocatedamongdifferentstockpointsforend-of-lifeproducts,disassembledbut notrecoveredparts,andrecoveredparts.Thesestockpointsdonotformaserial productionsystem,butarepointsfromwhichdemandcanbemetincaseamore value-addeditemisnotinstockanduponperformingthenecessarydemanufacturing operations.Wealsoinvestigateanalternativetestingschemeinadditiontotheexisting testingschemesconsideredinliterature,inwhichadiagnostictestcanbeconducted uponthedisassemblyoperation.Thisalternativeisoftenthepracticeinthecurrent demanufacturingindustry,andthroughnumericalstudy,weprovideconditionsunder whicheachtestingschemeyieldsmoreprots. 26

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CHAPTER3 ADEMANUFACTURER'SACQUISITIONPRICE,QUANTITYANDALLOCATION DECISIONSUNDERTHETRADE-OFFBETWEENENVIRONMENTALFEESAND OPERATIONALCOSTS 3.1BackgroundandMotivation Demanufacturingcanbedenedasthedisassemblyofend-of-lifeproductsin ordertorecoverreusablecomponentsorpartsontheseproducts,andrecyclethe restoftheproduct.Duetolucrativebusinessopportunitiesassociatedwithend-of-life products,theyarenowbeingpurchasedandsoldascommodities.Asthesemarkets forend-of-lifeproductsemerge,demanufacturersdeveloptheabilitytoexercisesome controlonthequantityandsometimesqualityoftheend-of-lifeproductsavailable tothefacility,andcanacquiremoreend-of-lifeproductsbyofferinghigheracquisition prices.Forinstance,acellularphonepartsdemanufacturerwouldbewillingtopay ahigheracquisitionpriceforanewermodelasthedemandforthepartsthatcanbe recoveredfromsuchaproductwouldbehigher.Therefore,indecidingonhowmuchto acquire,ademanufacturerneedstotakeintoaccounttheprice-sensitivityofthesupply ofend-of-lifeproducts.However,thisdecisionisalsotightlycoupledwiththeuncertainty associatedwiththerecoverabilityofthereusablepartthatcanberetrievedfromthe end-of-lifeproduct.Theacquisitionpricesetonend-of-lifeproductsdeterminesthe quantityofsupplythatislteredbytherandomdemanufacturingyieldrate,which,in turninuencesthestockingdecisions.Asaresult,thedemanufacturerhastoconsider therandomdemanufacturingyieldinplanningtheacquisitionandstockingdecisions. Sometypesofend-of-lifeproductscontainhazardousmaterials,andhence,may poseathreattotheenvironmentifnotstoredproperly.Manygovernmentsregulate storageofsuchtypesofend-of-lifeproductsinordertoreduceoreliminatepotential environmentaldamage.Amongdifferentregulationmechanismssuchassetting emissionlimitsoncertainairpollutants,requiringriskmanagementplansagainst spills,etc.,onecommonmechanismistochargeenvironmentalfees.Asaresult,the 27

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demanufacturermayhavetopayenvironmentalfeesforthestorageofsuchend-of-life products.Inthiscontext,ademanufacturermayavoidincurringenvironmentalfees bynotstoringanend-of-lifeproductuponreceipt,but,byimmediatelyperforming necessarydemanufacturingoperationstoretrievethereusableparts,andsellthehulkto amaterialrecycler.However,demanufacturingoperationssuchasdisassemblyofthe end-of-lifeproduct;extractionofareusablepartintheend-of-lifeproduct;and,recovery oftheextractedpartareusuallylaborintensive;hence,expensive.Whiletryingtoavoid environmentalfeesbyimmediatelyprocessingend-of-lifeproductsuponreceiptrather thanstoringthem,thedemanufacturermayrecovermorereusablepartsthanthereisa demandfor,andincurthecostsoftheseexpensiveoperationsunnecessarily.Inother words,thedemanufacturerfacesatrade-offbetweentheenvironmentalfeesandthe operationalcostsinplanningthedemanufacturingoperations. Anotherrelatedfactorinthiscontextthatimpactsthestockingdecisionishow quicklythecustomerswishtheirorderstobelled.Thatis,ifrecoveredpartsare stored,thenthecustomerdemandcanbesatisedimmediately.Whereasifend-of-life productsarestored,thentherecanbesomedelayinsatisfyingthedemanddueto thetimerequiredtoperformthedemanufacturingoperations.Ifthedemandcannot besatisedimmediately,thedemanufacturermighthavetocompensatethecustomer forthedelay.Insummary,thedecisionofwhattostockhingesonboththetrade-off betweentheenvironmentalfeesandtheoperationalcostsaswellasthecharacteristics ofthecustomers. Byincorporatingseveralimportantelementsfromourobservationsabove,our researchprovidesarealisticoperationalmodelforademanufacturer.Specically,we takeintoaccount:ithetrade-offbetweentheenvironmentalfeesandoperationalcosts, iiacquisitionprice-dependentsupplyofend-of-lifeproducts,iiistochasticdemandfor reusableparts,ivstochasticdemanufacturingyieldrate,andvcompensationdueto customerwait. 28

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3.2ProblemDenition Thedemanufacturerspeciesanacquisitionpricetobuyaparticulartypeof end-of-lifeproductfromanend-usermarket.Asingletypeofreusablepartmaybe recoveredfromthistypeofend-of-lifeproduct.Eachpartrequires,rst,adisassembly operationtoremovethepartfromtheend-of-lifeproduct,and,next,arecoveryoperation torestorethepartforreuseinasecondarymarket.Thedemanufacturermaykeepstock inthreedifferentforms:end-oflifeproducts,disassembledbutnotrecoveredparts,and recovereddisassembledandrecoveredparts,incurringtheassociatedenvironmental feesandoperationalcosts.Customerdemandforthereusablepartcanbesatised usinganyofthesestockpoints.ThedemanufacturingsystemisdepictedinFigure3-1: Figure3-1.Demanufacturingsystem. Supplyquantityoftheend-of-lifeproductsdependsontheacquisitionprice, C > 0 andisassumedtoincreaselinearlyintheacquisitionprice.Thatis,thesupplyquantity isgivenby a + bC a b > 0 where a and b denotethemarketpotentialandprice sensitivityofthesupplyfortheend-of-lifeproductofinterest,respectively. Tomodeltheuncertaintyassociatedwiththedemanufacturingyield,weuse continuousyield,referredtoas stochasticallyproportionalyield .Thatis,theyieldrateis denedonanintervalwithlowerandupperbounds,andisassumedtobeindependent ofthebatchsize.Inourwork,weassumethatthedemanufacturingyieldrate, Y ,is denedonaninterval [ y L y U ] 2 ,1] withacontinuousprobabilitydensityfunction pdf g y ,andacumulativedistributionfunction cdf G y .Eachacquiredend-of-life 29

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productistestedtoassesswhetherthereusablepartinitisrecoverable.Theunitcost ofthistestingoperationis c i .End-of-lifeproductsthatfailtopassthistestaresentto amaterialrecyclerfortheirmaterialcontent,collectingasalvagevalueof v forthe reusablepartitcontainsinadditiontoasalvagevalueof v h forthehulk.Thoseproducts thatpassthetestareallocatedamongthethreestockpointsinthesystemasfollows: 1.Someofthemarekeptintheend-of-lifeproductsstock.Thefacilityincursan environmentalfee, h p ,foreachunitofend-of-lifeproductkeptinstock.Notethat, duetothesettingweconsiderinourwork,theenvironmentalfeeisexpressedasa unitvariablecost,andnotasafunctionoftheweightorvolumeoftheproduct. 2.Someofthemmayundergoonlythedisassemblyoperationtoextractthe reusablepartsfromthem,ataunitcostof c d .Thesepartsarethenstoredin thedisassembledpartsstock,whereasthehulkcanbesoldtoamaterialrecycler foritsmaterialcontent,collectingthesalvagerevenueof v h 3.Therestmaygounderboththedisassemblyandtherecoveryoperations,by whichthereusablepartsarerstremovedfromtheend-of-lifeproductsandthen restoredforreuse.Theunitcostoftherecoveryoperationis c r .Thesepartsare storedintherecoveredpartsstockwhereasthehulkcanagainbesoldsoldtothe materialrecyclerforitsmaterialcontent. Inpractice,disassemblyandrecoveryoperationsarelaborintensiveandhaveto becarriedoutonaunitbasisInderfurthandLangella2006.Hence,weignoreall possiblexedcostsassociatedwithinitializingtheseoperations.Customerdemandfor thereusableparts, X ,isarandomvariablewithaknownprobabilitydensityfunction, f x ,andacumulativedistributionfunction, F x .Foreachunitofdemandsatised fromtherecoveredpartsstock,thedemanufacturercollectsarevenueof p perunit.If thedemandissatisedfromthedisassembledpartsstock,thedemanufacturercollects arevenueof p )]TJ/F39 11.9552 Tf 12.595 0 Td [(d perunit,i.e.,atadiscountedpricetocompensatethecustomer forthewaitingtimetoperformtherecoveryoperation.Finally,ifthedemandissatised fromtheend-of-lifeproductsstock,thedemanufacturercollectsarevenueof p )]TJ/F39 11.9552 Tf 12.632 0 Td [(e perunit,where e > d .Thedemanufacturerlosesrevenue p foreachunitofunsatised demandandincursaunitloss-of-goodwillcostof r perunit.Thepartsremaininginthe disassembledpartsstockandtherecoveredpartsstockcanbesoldfor v perunittothe 30

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materialrecyclerfortheirmaterialcontent.Similarly,theend-of-lifeproductsremaining instockcanbesoldfor v h + v perunittotherecycler.Ourdecisionvariablesinthis settingaretheunitacquisitioncost C tosetonend-of-lifeproducts;andthequantities ofend-of-lifeproductsacquiredtoallocateacrossthestocksof:end-of-lifeproducts P disassembledparts D ,andrecoveredparts R 3.3ModelDevelopmentandAnalysis First,wepresentourformulationforthesettingwhenthedemanufacturingyield rateismodeledexplicitlyasarandomvariable.Thedemanufacturermaystock end-of-lifeproducts,disassembledparts,andrecoveredparts.Werefertothismodel withstochasticdemanufacturingyieldrateandthreestockpointsas S3S .Wethen formulatesingle-stock-pointmodels,inwhichonlyonetypeofstockiskepteitherinthe theformofstockingonlyend-of-lifeproducts, S1S-P ;onlydisassembledparts, S1S-D ; oronlyrecoveredparts, S1S-R .Foreachofthesesettings,wealsoconsideraspecial casewherethesupplyofend-of-lifeproductsisindependentoftheacquisitionprice, andweadd IS tothefrontofthemodellabelstoindicatethis.Furthermore,wealso studybenchmarkmodelsinwhichthedemanufacturingyieldrateisdeterministic,and wereplacethe S with D inmodellabelstoindicatethis. 3.3.1StochasticYieldRate Thedemanufactureracquiresanumberofend-of-lifeproducts R + D + P ,and conductsatestontheseend-of-lifeproductstoidentifytheoneswithreusableparts thatcanberecovered.Theseend-of-lifeproducts,fromwhichreusablepartscanbe recovered,areallocatedacrossthestockpointsof:end-of-lifeproducts,disassembled parts,andrecoveredparts.Theallocationquantitiesarealsorandomvariables, YR YD ,and YP sincethedemanufacturingyieldrate, Y ,isarandomvariable.Thetotal acquisitionquantity, R + D + P ,canbeequatedtothesupply, a + bC ,fromwhichwe have: C = R + D + P )]TJ/F39 11.9552 Tf 11.543 0 Td [(a = b .Usingthisrelation,theexpectedprot, S 3 S R D P ,can beexpressedasinequation3-1: 31

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S3S : S 3 S R D P = Z y U y L Z yR 0 pxf x dx | {z } 3 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1. a + Z y R + D yR )]TJ/F39 11.9552 Tf 5.48 -9.683 Td [(pyR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r x )]TJ/F39 11.9552 Tf 11.955 0 Td [(yR f x dx | {z } 3 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1. b + Z y R + D + P y R + D )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(pyR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r yD + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r x )]TJ/F39 11.9552 Tf 11.955 0 Td [(yR )]TJ/F39 11.9552 Tf 11.955 0 Td [(yD f x dx | {z } 3 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1. c + Z 1 y R + D + P )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(pyR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r yD + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r yP f x dx | {z } 3 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1. d + Z y R + D + P 0 v )]TJ/F39 11.9552 Tf 5.48 -9.683 Td [(y R + D + P )]TJ/F39 11.9552 Tf 11.955 0 Td [(x f x dx | {z } 3 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1. e )]TJ/F30 11.9552 Tf 11.291 16.272 Td [(Z 1 y R + D + P r )]TJ/F39 11.9552 Tf 5.48 -9.683 Td [(x )]TJ/F39 11.9552 Tf 11.955 0 Td [(y R + D + P f x dx | {z } 3 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1. f )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p yP | {z } 3 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1. g )]TJ/F39 11.9552 Tf 11.291 0 Td [(c d y R + D | {z } 3 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1. h )]TJ/F39 11.9552 Tf 11.291 0 Td [(c r yR | {z } 3 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1. i + v )]TJ/F39 11.9552 Tf 11.955 0 Td [(y R + D + P | {z } 3 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1. j g y dy )]TJ/F22 11.9552 Tf 11.955 0 Td [( R + D + P )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b R + D + P | {z } 3 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1. k )]TJ/F39 11.9552 Tf 11.291 0 Td [(c i R + D + P | {z } 3 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1. ` + v h R + D + P | {z } 3 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1. m -1 Theouterintegralintheexpectedprotequation3-1givestherangeoverwhich thedemanufacturingyieldrateisdened.Theintegralsinsidecorrespondtothegross protsearned after theacquisitionofend-of-lifeproducts,testing,andstockallocation arecompleteduponcarryingthenecessarydemanufacturingoperations;and, after thedemand, x ,isobserved.Ifrecoveredpartsstockissufcienttosatisfythe entiredemand,revenuesareearnedfromthesales3-1.a.Ifrecoveredpartsstock isinsufcienttosatisfytheentiredemand,theexcessdemandcanbesatisedrst fromthedisassembledpartsstockaftercarryingouttherecoveryoperation3-1.b, andthenfromtheend-of-lifeproductstockaftercarryingoutboththedisassemblyand recoveryoperations3-1.c.Inallthesecases,revenuesareearnedfromthesales; costsareincurredforthenecessaryoperations.Ifthedemandexceedsthetotal stockinthesystem,onlysomeportionofthedemandcanbesatised3-1.d,and penaltycostsareincurredfortheunsatisedportionofthedemand3-1.f.Ifthere areleft-overitems,thenthesearesoldforthematerialvalueofthereusableparts, collectingthesalvagerevenues3-1.e.Thefacilityincurstheenvironmentalfees 32

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duetostoringend-of-lifeproducts3-1.g,costsoftheoperations3-1.hand3-1.i performedontheend-of-lifeproductswhichpassthetestbeforetheacquiredstockis allocatedamongthestockpoints.Salvagerevenuesareearnedfromtheend-of-life productsthatfailthetest3-1.j.Thetermsin3-1.k,3-1. ` ,and3-1.mcorrespond tothe:totalacquisitioncostoftheend-of-lifeproducts,costofdiagnostictests,and salvagerevenueearnedfromthehulkofallacquiredend-of-lifeproducts,respectively. Derivationsofexpectedprotfunctionsforthesingle-stock-pointmodelsunder stochasticdemanufacturingyieldratearestraightforwardandareprovidedinequations 3-2,3-3,and3-4below: S1S-P : S 1 S )]TJ/F40 7.9701 Tf 6.586 0 Td [(P P = Z y U y L Z yP 0 p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.956 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r xf x dx + Z 1 yP p )]TJ/F39 11.9552 Tf 11.956 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r yPf x dx + Z yP 0 v yP )]TJ/F39 11.9552 Tf 11.955 0 Td [(x f x dx )]TJ/F30 11.9552 Tf 11.956 16.273 Td [(Z 1 yP r x )]TJ/F39 11.9552 Tf 11.955 0 Td [(yP f x dx )]TJ/F39 11.9552 Tf 11.956 0 Td [(h p yP + v )]TJ/F39 11.9552 Tf 11.955 0 Td [(y P g y dy )]TJ/F39 11.9552 Tf 11.955 0 Td [(P P )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i P + v h P -2 S1S-D : S 1 S )]TJ/F40 7.9701 Tf 6.586 0 Td [(D D = Z y U y L Z yD 0 p )]TJ/F39 11.9552 Tf 11.956 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r xf x dx + Z 1 yD p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r yDf x dx + Z yD 0 v yD )]TJ/F39 11.9552 Tf 11.955 0 Td [(x f x dx )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z 1 yD r x )]TJ/F39 11.9552 Tf 11.955 0 Td [(yD f x dx )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d yD + v )]TJ/F39 11.9552 Tf 11.955 0 Td [(y D g y dy )]TJ/F39 11.9552 Tf 11.955 0 Td [(D D )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i D + v h D -3 S1S-R : S 1 S )]TJ/F40 7.9701 Tf 6.586 0 Td [(R R = Z y U y L Z yR 0 pxf x dx + Z 1 yR pyRf x dx + Z yR 0 v yR )]TJ/F39 11.9552 Tf 11.955 0 Td [(x f x dx )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z 1 yR r x )]TJ/F39 11.9552 Tf 11.955 0 Td [(yR f x dx )]TJ/F22 11.9552 Tf 11.955 0 Td [( c d + c r yR + v )]TJ/F39 11.9552 Tf 11.955 0 Td [(y R g y dy )]TJ/F39 11.9552 Tf 11.955 0 Td [(R R )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b )]TJ/F39 11.9552 Tf 11.956 0 Td [(c i R + v h R -4 Whenthesupplyofend-of-lifeproductsisindependentoftheacquisitionprice, expectedprotsfor ISS3S canbederivedsimilartotheacquisition-price-dependent casebyreplacingtheacquisitioncostterm R + D + P )]TJ/F39 11.9552 Tf 12.69 0 Td [(a = b inequation3-1.k withtheacquisitionpriceparameter, c .Theexpectedprotfunctioninequation3-1is strictlyconcaveasstatedinTheorem3.1: 33

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Theorem3.1. S 3 S R D P isstrictlyconcavein R D and P ifandonlyifthe followingconditionshold:i p )]TJ/F39 11.9552 Tf 12.976 0 Td [(e )]TJ/F39 11.9552 Tf 12.975 0 Td [(c d )]TJ/F39 11.9552 Tf 12.975 0 Td [(c r > v )]TJ/F39 11.9552 Tf 12.975 0 Td [(r ,i.e.,satisfyingthereusable partdemandfromthestockofend-of-lifeproductsismoreprotablethansalvaging thepart,earningthesalvagerevenueandincurringthepenaltyforthelostsale,ii e + c d + c r > d + c r ,i.e.,themarginalcostofpostponingthedisassemblyoperationis greaterthanthemarginalcostofpostponingtherecoveryoperation,andiii d + c r > 0 i.e.,themarginalcostofpostponingtherecoveryoperationispositive. Proof. FirstorderoptimalityconditionsandtheHessianmatrix, H ,are: S 3 S 0 R = )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z y U y L p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r yF y R + D + P g y dy )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z y U y L e )]TJ/F39 11.9552 Tf 11.955 0 Td [(d + c d yF y R + D g y dy )]TJ/F30 11.9552 Tf 11.956 16.273 Td [(Z y U y L d + c r yF yR g y dy )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = b + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r E [ Y ] )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v h + v + a = b S 3 S 0 D = )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z y U y L p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r yF y R + D + P g y dy )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z y U y L e )]TJ/F39 11.9552 Tf 11.955 0 Td [(d + c d yF y R + D g y dy )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = b + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r E [ Y ] )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v h + v + a = b S 3 S 0 P = )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z y U y L p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.956 0 Td [(c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r yF y R + D + P g y dy )]TJ/F22 11.9552 Tf 11.956 0 Td [(2 R + D + P = b + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.956 0 Td [(c d )]TJ/F39 11.9552 Tf 11.956 0 Td [(c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p E [ Y ] )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v h + v + a = b H = )]TJ/F23 7.9701 Tf 6.587 0 Td [( A + B + C )]TJ/F23 7.9701 Tf 6.587 0 Td [( A + B )]TJ/F40 7.9701 Tf 6.587 0 Td [(A )]TJ/F23 7.9701 Tf 6.587 0 Td [( A + B )]TJ/F23 7.9701 Tf 6.587 0 Td [( A + B )]TJ/F40 7.9701 Tf 6.587 0 Td [(A )]TJ/F40 7.9701 Tf 6.587 0 Td [(A )]TJ/F40 7.9701 Tf 6.587 0 Td [(A )]TJ/F40 7.9701 Tf 6.587 0 Td [(A where, E [ ] istheexpectedvalueoftheargument, A = R y U y L p )]TJ/F39 11.9552 Tf 12.813 0 Td [(e )]TJ/F39 11.9552 Tf 12.813 0 Td [(c d )]TJ/F39 11.9552 Tf 12.813 0 Td [(c r )]TJ/F39 11.9552 Tf 12.813 0 Td [(v + r y 2 f y R + D + P g y dy +2 = b B = R y U y L e )]TJ/F39 11.9552 Tf 12.921 0 Td [(d + c d y 2 f y R + D g y dy and C = R y U y L d + c r y 2 f yR g y dy H isnegativedeniteifandonlyifthethe termsinparenthesesinsidetheintegralsfor A B C arepositive.Wefurthernotethat conditionsiiandiiiareunnecessaryunderthenon-trivialcoststructureandare merelypresentedforthesakeofcompleteness. 34

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Underconcavity,KKTconditionsaresufcient,and,sincetheconstraintsi.e., non-negativityoftheacquisitionquantitiesarelinear,theyarealsonecessaryfor optimality.Inaddition,sincetheexpectedprotfunctionisstrictlyconcave,theoptimum isunique.Therefore,thetriple, R D P ,maximizing S 3 S R D P canbedetermined usingtherstorderconditions.However,dependingonthedemanddistribution,explicit solutionsmaynotbeavailable.Furthermore,duetothenon-negativityconstraintsand the cdf sinvolved,therstorderconditionsmaynotyieldafeasiblesolution.Hence, wecheckifthestationarypointobtainedusingtherstorderconditionsisfeasible.If thestationarypointisfeasible,thenitistheuniqueoptimum,otherwise,afeasiblepoint thatsatisestheKKTconditionsissearchedforbycheckingallpossiblecases,i.e., combinationsofdualvariablesatnegativeandzerolevelsassummarizedinTable3-1: Table3-1.S3S:KKTcasesunderstochasticdemanufacturingyieldrateandgeneral demanddistribution. Case RDP S 3 S 0 R S 3 S 0 D S 3 S 0 P Equationstosolve C1 0 0 0 000 E [ YF YR ]= dE [ Y ] = d + c r E [ YF Y R + D ]= e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p E [ Y ] = e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r E [ YF Y R + D + P ] +2 R + D + P = b = p )]TJ/F39 9.9626 Tf 9.963 0 Td [(h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v h + v + a = b C20 0 0 < 0 00 E [ YF YD ]= e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p E [ Y ] = e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r E [ YF Y D + P ]+2 D + P = b = p )]TJ/F39 9.9626 Tf 9.963 0 Td [(h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v h + v + a = b C30 0 0 < 0 0 < 0 p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r E [ YF YD ]+2 D = b = p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v h + v + a = b C4 0 0 0 0 < 0 0 E [ YF YR ]= e + h p E [ Y ] = e + c d + c r p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r E [ YF Y R + P ]+2 R + P = b = p )]TJ/F39 9.9626 Tf 9.963 0 Td [(h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v h + v + a = b C5 0 000 < 0 < 0 p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r E [ YF YR ]+2 R = b = p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v h + v + a = b C6 0 0 000 < 0 E [ YF YR ]= dE [ Y ] = d + c r p )]TJ/F39 9.9626 Tf 9.962 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r E [ YF Y R + D ]+2 R + D = b = p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v h + v + a = b C700 0 < 0 < 0 0 p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r E [ YF YP ]+2 P = b = p )]TJ/F39 9.9626 Tf 9.963 0 Td [(h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v h + v + a = b C8000 < 0 < 0 < 0 NotethateachKKTcaseinTable3-1prescribesaparticularstockingstructure. Azeroacquisitionquantitymeansnotstockingatthecorrespondingstockpoint.For 35

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instance,C5correspondstostockingonlyrecoveredpartswhereasC7toend-of-life productsonly.Intheformercase,allacquiredend-of-lifeproductscontainingrecoverable partsgothroughbothdisassemblyandrecoveryoperationsimmediately.Hence, noenvironmentalfeesordiscountsareincurred.Inthelattercase,nodemanufacturing operationsareperformeduntilademandoccurs.Therefore,thefacilitysavesoperational costsbypostponingalloperations,butincursenvironmentalfeesandthelargest discountduetomakingcustomerwaitforbothoperations.SinceC1givestheunique optimizeroftheexpectedprotfunction,thethree-stock-pointmodelisthebeststocking structureaslongasthestockquantitiesarefeasible.Byinsertingtheequationstosolve fordecisionvariablesintotherelevantrstorderconditions,itcanbeseenthatsomeof theKKTcasescanbeeliminatedsincetheydonotsatisfytheKKTconditions: Corollary3.1.1. TheoptimalstockingstructureisgivenbyC1,i.e.,stockingendof-lifeproducts,disassembledparts,andrecoveredparts,ifthestockquantitiesare feasible.Otherwise,theoptimalstockingstructureisgivenbyeitherof:C4,C5,or,C6, i.e.,stockingend-of-lifeproductsandrecoveredparts,stockingonlyrecoveredparts,or stockingdisassembledpartsandrecoveredparts,respectively. Oncetheoptimalvaluesoftheacquisitionquantitiesarefound,itisstraightforward toobtaintheoptimalacquisitionpriceusingtheidentitybetweenthesupplyand acquisitionquantities.WefurthernotethatexplicitsolutionstoKKTcasescanbe obtainedwhenthedemanddistributionisuniformassummarizedinTable3-2: Table3-2.S3S:KKTcasesunderstochasticdemanufacturingyieldrateand uniform U demand. Case RDP S 3 S 0 R S 3 S 0 D S 3 S 0 P Equationstosolve C1 0 0 0 000 F R = dE [ Y ] = d + c r E [ Y 2 ] F R + D = e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p E [ Y ] = e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d E [ Y 2 ] F R + D + P = p )]TJ/F40 6.9738 Tf 6.226 0 Td [(h p )]TJ/F40 6.9738 Tf 6.227 0 Td [(e )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r E [ Y ] )]TJ/F40 6.9738 Tf 6.227 0 Td [(c i + v h + v + a = b p )]TJ/F40 6.9738 Tf 6.227 0 Td [(e )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r E [ Y 2 ]+2 U = b C4 0 0 0 0 < 0 0 F R = e + h p E [ Y ] = e + c d + c r E [ Y 2 ] F R + P = p )]TJ/F40 6.9738 Tf 6.227 0 Td [(h p )]TJ/F40 6.9738 Tf 6.227 0 Td [(e )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r E [ Y ] )]TJ/F40 6.9738 Tf 6.226 0 Td [(c i + v h + v + a = b p )]TJ/F40 6.9738 Tf 6.226 0 Td [(e )]TJ/F40 6.9738 Tf 6.226 0 Td [(c d )]TJ/F40 6.9738 Tf 6.226 0 Td [(c r )]TJ/F40 6.9738 Tf 6.226 0 Td [(v + r E [ Y 2 ]+2 U = b C5 0 000 < 0 < 0 F R = p )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r E [ Y ] )]TJ/F40 6.9738 Tf 6.226 0 Td [(c i + v h + v + a = b p )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r E [ Y 2 ]+2 U = b C6 0 0 000 < 0 F R = dE [ Y ] = d + c r E [ Y 2 ] F R + D = p )]TJ/F40 6.9738 Tf 6.227 0 Td [(d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r E [ Y ] )]TJ/F40 6.9738 Tf 6.226 0 Td [(c i + v h + v + a = b p )]TJ/F40 6.9738 Tf 6.227 0 Td [(d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.226 0 Td [(v + r E [ Y 2 ]+2 U = b 36

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Furthermore,wederivelowerboundsontheexpecteddemanufacturingyieldrate thatarenecessaryforC1,C4,C5,andC6,aswellasthesingle-stock-pointmodels. Usingalowerbound,itispossibletoobtainarangeofdemanufacturingyieldratesover whichthebusinessisprotabletooperate,whichmayalsohelpthedemanufacturer selectacquisitionsourcesaccordingly.AppendixAlistslowerboundsobtainedfor allmodels,andincludesademonstrationofthelowerboundsatasetofparameters. Corollary3.1.2provideslowerboundsunder S3S : Corollary3.1.2. Let = c i )]TJ/F39 11.9552 Tf 13.172 0 Td [(v )]TJ/F39 11.9552 Tf 13.172 0 Td [(v h and = p )]TJ/F39 11.9552 Tf 13.172 0 Td [(v + r .Lowerbounds on E [ Y ] forS3SinKKTcasesC1andC4,C5,andC6aregivenby:i E [ Y ] max f 0, + a = b = )]TJ/F39 11.9552 Tf 11.55 0 Td [(h p )]TJ/F39 11.9552 Tf 11.55 0 Td [(e )]TJ/F39 11.9552 Tf 11.549 0 Td [(c d )]TJ/F39 11.9552 Tf 11.549 0 Td [(c r g ,ii E [ Y ] max f 0, + a = b = )]TJ/F39 11.9552 Tf 11.549 0 Td [(c d )]TJ/F39 11.9552 Tf 11.549 0 Td [(c r g ,and iii E [ Y ] max f 0, + a = b = )]TJ/F39 11.9552 Tf 12.304 0 Td [(d )]TJ/F39 11.9552 Tf 12.304 0 Td [(c d )]TJ/F39 11.9552 Tf 12.304 0 Td [(c r g ,respectively.When E [ Y ] satisesi, italsosatisesiiandiii.Furthermore,lowerboundson E [ Y ] for S1S-P and S1S-R and S1S-D aregivenbyiandii,respectively.Lowerboundson E [ Y ] arenecessary forC1,C4,C5,andC6toholdundertheconcavityconditionsgiveninTheorem3.1. Proof. WewillderivetheboundinC1andomittherestofthecasessincetheproofs aresimilar.KKTequationtosolveinC1canberewrittenas )]TJ/F39 11.9552 Tf 10.778 0 Td [(e )]TJ/F39 11.9552 Tf 10.779 0 Td [(c d )]TJ/F39 11.9552 Tf 10.778 0 Td [(c r E [ YF Y R + D + P ]+2 C = )]TJ/F39 11.9552 Tf 10.339 0 Td [(e )]TJ/F39 11.9552 Tf 10.339 0 Td [(h p )]TJ/F39 11.9552 Tf 10.339 0 Td [(c d )]TJ/F39 11.9552 Tf 10.34 0 Td [(c r E [ Y ] )]TJ/F22 11.9552 Tf 10.34 0 Td [( + a = b .Sincelefthandsideofthisequation isnon-negative,somustbetherighthandside,fromwhichtheprooffollows. 3.3.2DeterministicYieldRate Inthissection,weprovidebenchmarkmodelswherethedemanufacturingyield rateismodeledasadeterministicparameter.Let y denotethedeterministicyieldrate, and P D R correspondtothestocklevelsofend-of-lifeproducts,disassembledparts, andrecoveredparts,respectively.Hence,atotalof R + D + P = y end-of-lifeproducts areacquired.Theacquisitionpricecanbeexpressedintermsofthestockquantitiesas C = )]TJ/F22 11.9552 Tf 5.479 -9.683 Td [( R + D + P = y )]TJ/F39 11.9552 Tf 10.254 0 Td [(a = b .Theexpectedprot, D 3 S canbewrittenasinequation3-5: D3S : D 3 S R D P = Z R 0 pxf x dx + Z R + D R )]TJ/F39 11.9552 Tf 5.479 -9.683 Td [(pR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r x )]TJ/F39 11.9552 Tf 11.955 0 Td [(R f x dx 37

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+ Z R + D + P R + D )]TJ/F39 11.9552 Tf 5.48 -9.683 Td [(pR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r D + p )]TJ/F39 11.9552 Tf 11.956 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r x )]TJ/F39 11.9552 Tf 11.955 0 Td [(R )]TJ/F39 11.9552 Tf 11.956 0 Td [(D f x dx + Z 1 R + D + P )]TJ/F39 11.9552 Tf 5.479 -9.683 Td [(pR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r D + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.956 0 Td [(c d )]TJ/F39 11.9552 Tf 11.956 0 Td [(c r P f x dx + Z R + D + P 0 v R + D + P )]TJ/F39 11.9552 Tf 11.955 0 Td [(x f x dx )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z 1 R + D + P r x )]TJ/F39 11.9552 Tf 11.955 0 Td [(R )]TJ/F39 11.9552 Tf 11.956 0 Td [(D )]TJ/F39 11.9552 Tf 11.955 0 Td [(P f x dx )]TJ/F30 11.9552 Tf 11.955 9.683 Td [()]TJ/F22 11.9552 Tf 5.479 -9.683 Td [( R + D + P = y )]TJ/F22 11.9552 Tf 10.959 -9.683 Td [( R + D + P = y )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p P )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i R + D + P = y )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d R + D )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r R + )]TJ/F39 11.9552 Tf 5.479 -9.683 Td [(v h + v )]TJ/F39 11.9552 Tf 11.955 0 Td [(y R + D + P = y -5 Concavityconditionsandthecorrespondingprooffor D 3 S in3-5arethesameas thosefor S 3 S ,andhence,areomittedfromdiscussion.Firstorderconditionsarebelow: D 3 S 0 R = )]TJ/F22 11.9552 Tf 11.956 0 Td [( p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r )]TJ/F39 11.9552 Tf 11.956 0 Td [(v + r F R + D + P )]TJ/F22 11.9552 Tf 11.955 0 Td [( e )]TJ/F39 11.9552 Tf 11.955 0 Td [(d + c d F R + D )]TJ/F22 11.9552 Tf 11.956 0 Td [( d + c r F R )]TJ/F22 11.9552 Tf 11.956 0 Td [(2 R + D + P = by 2 + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F22 11.9552 Tf 11.955 0 Td [( c i )]TJ/F39 11.9552 Tf 11.955 0 Td [(v h )]TJ/F39 11.9552 Tf 11.955 0 Td [(v )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b = y D 3 S 0 D = )]TJ/F22 11.9552 Tf 11.955 0 Td [( p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r F R + D + P )]TJ/F22 11.9552 Tf 11.955 0 Td [( e )]TJ/F39 11.9552 Tf 11.955 0 Td [(d + c d F R + D )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = by 2 + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F22 11.9552 Tf 11.955 0 Td [( c i )]TJ/F39 11.9552 Tf 11.955 0 Td [(v h )]TJ/F39 11.9552 Tf 11.955 0 Td [(v )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b = y D 3 S 0 P = )]TJ/F22 11.9552 Tf 11.955 0 Td [( p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.956 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r F R + D + P )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = by 2 + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r )]TJ/F39 11.9552 Tf 11.956 0 Td [(v + r )]TJ/F22 11.9552 Tf 11.955 0 Td [( c i )]TJ/F39 11.9552 Tf 11.955 0 Td [(v h )]TJ/F39 11.9552 Tf 11.956 0 Td [(v )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b = y Thetriple R D P maximizing D 3 S R D P canbedeterminedusingtherst orderconditionsduetostrictconcavityasstatedinCorollary3.1.3: Corollary3.1.3. Optimalstockingquantitiesforthedisassembledandrecovered partsunderdeterministicdemanufacturingyieldratearedeterminedbythefollowing probabilitiesofnotstockingout: F R = d = d + c r and F R + D = e )]TJ/F39 11.9552 Tf 12.158 0 Td [(d + h p = e )]TJ/F39 11.9552 Tf -451.55 -23.908 Td [(d + c d ,andtheoptimalstockingquantityoftheend-of-lifeproductsisthesolutionto D 3 S 0 P =0 ,providedthattheobjectivefunctionisstrictlyconcaveanditsstationary pointisfeasible. Similarto S3S ,duetothenon-negativityconstraintsand cdf sinvolved,therst orderconditionsmaynotyieldafeasiblesolution.Therefore,asimilarapproach 38

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utilizingKKTconditionscanbeusedtondtheoptimalsolution.Theimplicationsof Corollary3.1.1holdfor D3S aswell,andhence,willnotberepeated.Tables3-3and3-4 presentKKTcasesundergeneralanduniformdemanddistributionsrespectively: Table3-3.D3S:KKTcasesunderdeterministicdemanufacturingyieldrateandgeneral demanddistribution. Case RDP D 3 S 0 R D 3 S 0 D D 3 S 0 P Equationstosolve C1 0 0 0 000 F R = d = d + c r F R + D = e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p = e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d D 3 S 0 P =0 C20 0 0 < 0 00 F D = e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p = e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d D 3 S 0 P =0 C30 0 0 < 0 0 < 0 D 3 S 0 D =0 C4 0 0 0 0 < 0 0 F R = e + h p = e + c d + c r D 3 S 0 P =0 C5 0 000 < 0 < 0 D 3 S 0 R =0 C6 0 0 000 < 0 F R = d = d + c r D 3 S 0 D =0 C700 0 < 0 < 0 0 D 3 S 0 P =0 C8000 < 0 < 0 < 0 Table3-4.D3S:KKTcasesunderdeterministicdemanufacturingyieldrateanduniform U demand. Case RDP D 3 S 0 R D 3 S 0 D D 3 S 0 P Equationstosolve C1 0 0 0 000 F R = d = d + c r F R + D = e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p = e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d F R + D + P = p )]TJ/F40 6.9738 Tf 6.226 0 Td [(h p )]TJ/F40 6.9738 Tf 6.227 0 Td [(e )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r )]TJ/F23 6.9738 Tf 6.227 0 Td [( c i )]TJ/F40 6.9738 Tf 6.226 0 Td [(v h )]TJ/F40 6.9738 Tf 6.227 0 Td [(v )]TJ/F40 6.9738 Tf 6.226 0 Td [(a = b = y p )]TJ/F40 6.9738 Tf 6.227 0 Td [(e )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r +2 U = by 2 C20 0 0 < 0 00 F D = e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p = e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d F D + P = p )]TJ/F40 6.9738 Tf 6.226 0 Td [(h p )]TJ/F40 6.9738 Tf 6.227 0 Td [(e )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r )]TJ/F23 6.9738 Tf 6.227 0 Td [( c i )]TJ/F40 6.9738 Tf 6.226 0 Td [(v h )]TJ/F40 6.9738 Tf 6.227 0 Td [(v )]TJ/F40 6.9738 Tf 6.226 0 Td [(a = b = y p )]TJ/F40 6.9738 Tf 6.227 0 Td [(e )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r +2 U = by 2 C30 0 0 < 0 0 < 0 F D = p )]TJ/F40 6.9738 Tf 6.227 0 Td [(d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r )]TJ/F23 6.9738 Tf 6.227 0 Td [( c i )]TJ/F40 6.9738 Tf 6.226 0 Td [(v h )]TJ/F40 6.9738 Tf 6.227 0 Td [(v )]TJ/F40 6.9738 Tf 6.226 0 Td [(a = b = y p )]TJ/F40 6.9738 Tf 6.226 0 Td [(d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r +2 U = by 2 C4 0 0 0 0 < 0 0 F R = e + h p = e + c d + c r F R + P = p )]TJ/F40 6.9738 Tf 6.226 0 Td [(h p )]TJ/F40 6.9738 Tf 6.227 0 Td [(e )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r )]TJ/F23 6.9738 Tf 6.227 0 Td [( c i )]TJ/F40 6.9738 Tf 6.226 0 Td [(v h )]TJ/F40 6.9738 Tf 6.227 0 Td [(v )]TJ/F40 6.9738 Tf 6.226 0 Td [(a = b = y p )]TJ/F40 6.9738 Tf 6.227 0 Td [(e )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r +2 U = by 2 C5 0 000 < 0 < 0 F R = p )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r )]TJ/F23 6.9738 Tf 6.227 0 Td [( c i )]TJ/F40 6.9738 Tf 6.226 0 Td [(v h )]TJ/F40 6.9738 Tf 6.227 0 Td [(v )]TJ/F40 6.9738 Tf 6.226 0 Td [(a = b = y p )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r +2 U = by 2 C6 0 0 000 < 0 F R = d = d + c r F R + D = p )]TJ/F40 6.9738 Tf 6.227 0 Td [(d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r )]TJ/F23 6.9738 Tf 6.227 0 Td [( c i )]TJ/F40 6.9738 Tf 6.226 0 Td [(v h )]TJ/F40 6.9738 Tf 6.227 0 Td [(v )]TJ/F40 6.9738 Tf 6.226 0 Td [(a = b = y p )]TJ/F40 6.9738 Tf 6.226 0 Td [(d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r +2 U = by 2 C700 0 < 0 < 0 0 F P = p )]TJ/F40 6.9738 Tf 6.226 0 Td [(h p )]TJ/F40 6.9738 Tf 6.227 0 Td [(e )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r )]TJ/F23 6.9738 Tf 6.227 0 Td [( c i )]TJ/F40 6.9738 Tf 6.226 0 Td [(v h )]TJ/F40 6.9738 Tf 6.227 0 Td [(v )]TJ/F40 6.9738 Tf 6.226 0 Td [(a = b = y p )]TJ/F40 6.9738 Tf 6.227 0 Td [(e )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(v + r +2 U = by 2 C8000 < 0 < 0 < 0 Itisstraightforwardtondtheoptimalacquisitionpriceusingtherelationbetween therequiredacquisitionquantityandthesupply.Furthermore,similartothecasewhen thedemanufacturingyieldrateisstochastic,lowerboundsonthedemanufacturing yieldratecanbeobtained,albeitexplicitlyusingtheKKTequations.Thelowerbounds 39

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arethesameasinCorollary3.1.2exceptthattheseboundsareforthedeterministic demanufacturingyieldrate,andhenceareomittedfromdiscussion.Derivationsrelated tothesingle-stock-pointmodels, D1S-P D1S-D ,and D1S-R ,arestraightforwardandare presentedinAppendixB.Whenthesupplyquantityisindependentoftheacquisition price,wecanderiveexplicitprobabilitiesofnotstockingoutforthecorrespondingmodel, ISD3S seealsoAppendixB,asstatedinProposition3.1. Proposition3.1. Optimalstockingquantitiesforend-of-lifeproducts,disassembledparts,andrecoveredpartsunderdeterministicdemanufacturingyieldrateand acquisition-price-independentend-of-lifeproductsupply,aredeterminedbythefollowingprobabilitiesofnotstockingout:i F R = d = d + c r ,ii F R + D = e )]TJ/F39 11.9552 Tf 11.338 0 Td [(d + h p = e )]TJ/F39 11.9552 Tf 11.338 0 Td [(d + c d ,andiii F R + D + P = p )]TJ/F39 11.9552 Tf 11.338 0 Td [(h p )]TJ/F39 11.9552 Tf 11.338 0 Td [(e )]TJ/F39 11.9552 Tf 11.338 0 Td [(c d )]TJ/F39 11.9552 Tf 11.338 0 Td [(c r )]TJ/F39 11.9552 Tf 11.338 0 Td [(v + r )]TJ/F22 11.9552 Tf 11.338 0 Td [( c + c i )]TJ/F39 11.9552 Tf -458.702 -23.908 Td [(v h )]TJ/F39 11.9552 Tf 12.184 0 Td [(v = y = p )]TJ/F39 11.9552 Tf 12.184 0 Td [(e )]TJ/F39 11.9552 Tf 12.184 0 Td [(c d )]TJ/F39 11.9552 Tf 12.184 0 Td [(c r )]TJ/F39 11.9552 Tf 12.184 0 Td [(v + r ,providedthatobjectivefunctionisstrictlyconcave anditsstationarypointisfeasible. ProbabilitiesinProposition3.1provideinsightintothenewsvendorstructureof themodel.Ini,thecostofunderstockingisthediscount, d ,thatcouldbesavedif therewasonemoreunitofrecoveredpartinstock.Thecostofoverstockingisthe costofrecoveryoperation, c r ,thatcouldbesavedifadisassembledpartwasnot recovered.Theprobabilityofnotstockingoutiniiisgivenforthecombinedstocks ofthedisassembledpartsandtherecoveredparts.Thecostofunderstockingisthe discountsavings e )]TJ/F39 11.9552 Tf 12.759 0 Td [(d plustheenvironmentalfee, h p ,thatcouldbesavedifthere wasonemoreunitofdisassembledpartinstock.Thecostofoverstockingisthecost ofdisassemblyoperation, c d ,minustheenvironmentalfee, h p ,thatcouldbesavedif therewasonelessdisassembledpartinstock.Iniii,theprobabilityofnotstockingout isgivenforallofthestockpointscombined.Thecostofunderstockingisthenetprot thatcouldbeearnediftherewasonemoreunitintheend-of-lifeproductsstock.The overstockingcostisgivenbytheacquisitioncostplustheenvironmentalfeepaidminus therevenuesearnedbysellingtheparttothematerialrecycler. 40

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3.3.3AnalysisoftheKKTEquations Acquisitionorallocationquantitiesaredeterminedbytheprobabilitiesofnot stockingout.Therefore,effectsofparametersonthesequantitiescanbedetermined byexaminingtheirstructuresseeAppendixCforexplicitsolutionstoprobabilities ofnotstockingoutunderuniformlydistributeddemand.Table3-5summarizes howtheacquisitionunderstochasticdemanufacturingyieldrate,ortheallocation underdeterministicdemanufacturingyieldratequantitychangeswithincreasing valuesofproblemparametersunderthree-stock-pointlabeledas 3S forbrevityand single-stock-pointlabeledas 1S models: Table3-5.Sensitivityofacquisitionorallocationquantitiestosystemparameters. 3S:C13S:C43S:C53S:C6 1S-P1S-D1S-R Parameter RR + DR + D + PRR + PRRR + DPDR a """"""" b """"""" c ####### c i ####### c d ######### c r ########## h p "#"## d "#"## e "#"## r """"""" v h """"""" v """"""" Acquisition-price-dependentsupplymodelsonly. Acquisition-price-independentsupplymodelsonly. KKTcasesC3,C5,andC7arestructurallythesameasthosefor 1S-D 1S-R ,and 1S-P ,respectively.Therefore,itispossibleforthesingle-stock-pointmodelstoacquire andallocatethesametotalquantitiesas 3S dependingonthefeasibilityofC1,i.e., whethertheprobabilitiesofnotstockingoutarefeasibleunderthiscase.Ingeneral, totalacquisitionandallocationquantitiesof 3S areupperboundsontherespective quantitiesof 1S .Thisisbecause, 3S isabletoaffordhigherquantitiesofend-of-life productsduetohavingadditionaldegreesoffreedomwiththreestockpoints,andhence isabletobalancethecostsefcientlytogeneratehigherprots.IfC1holds,then 3S 41

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yieldslargeracquisitionandallocationquantitiesthan 1S exceptfor 1S-P .Sincethe probabilityofnotstockingoutinthesystemi.e., F R + D + P isthesameforboth models,thetotalacquisitionorallocationquantitiesofbothmodelsarethesameifC1 holdsunder 3S .ProbabilityofnotstockingoutinthesystemunderC1andC4equals tothatunder 1S-P ;underC6equaltothatunder 1S-D ;andunderC5equaltothat under 1S-R .Therefore, 3S andoneofthesingle-stock-pointmodelsalwayshavethe sametotalacquisitionandallocationquantities.Furthermore,recallthatacquisition priceisinterrelatedwiththetotalacquisitionquantityunderacquisition-price-dependent supply.Accordingly,theacquisitionpricedependsonalloftheparametersthatthe acquisitionortheallocationquantityisaffectedfrom,aswellasthesupplyparameters. 3.4ComputationalExperiments Wepreviouslypresentedthesensitivityofacquisitionandstockallocationquantities aswellasacquisitionpricetomodelparametersbasedonthestructureoftheKKT equationsin x 3.3.3.Inthissection,wereportthesensitivityoftheprotstolevelsof somemodelparametersinordertoassesstheimpactsofpostponingversusprioritizing thedisassemblyandrecoveryoperations. Inournumericalstudy,wesettheinterceptofthesupplyfunction, a ,at10%ofthe meandemand.Theslopeofthesupplyfunction, b ,isvariedattwolevelsoftheintercept %and30%.Unitacquisitioncost, c ,issetat30%oftheunitsellingpriceforthe experimentswhenthesupplyofend-of-lifeproductsdoesnotdependontheacquisition price. Wegeneratethevaluesofthecostparametersbasedonthesellingprice, p ,of arecoveredpartsincethisprovidesrealisticcostingschemeswithseveralrevenue levels.Basedonourexperiments,weobservedthatthesellingpriceonlyaffectsthe protsearnedanditdoesnotaffectthehowthemodelsbehave.Therefore,wexthe sellingpriceat50,anduseitasabaseforthevaluesoftheothercostparametersviaa percentage. 42

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Inreality,somediagnostictestsaremorecostlycomparedtoothers,andhence,to coverawiderange,weagainsetlowandhighlevelsforthecostoftestinganacquired end-of-lifeproduct, c i ,.1%and0.5%. Inordertokeeptheexperimentsatamanageablelevel,themeandemanufacturing yieldrate, Y ,isvariedatsixlevels.70,0.75, ... ,0.95.Weconsidervedifferent rangesforeachmeandemanufacturingyieldratebasedonthecoefcientofvariation, CV Y ,.0061,0.0122,0.0182,0.0243,0.0304.Hence,weareabletocomparesystem performances:iamongexperimentsatthesamemeandemanufacturingyieldrate wheretherangeoftheyieldrateisvaried,andiiacrossexperimentsatdifferentmean demanufacturingyieldrateswherethecoefcientofcorrelationisheldconstant.Forthe deterministicdemanufacturingyieldratecases,theabovemeanratesaretakenasthe deterministiccounterparts. Inordertocapturetheimpactoftheenvironmentalfee, h p ,ontheacquisitionand stockallocationdecisions,weincludeabenchmarkcasewithnoenvironmentalfee alongwiththreeothersettingswithlow,medium,andhighlevels.5%,3%,and6%.. Tocovermorerealisticanddiversecases,weconsiderthreedifferentweighingschemes fortheoperationalcosts:disassemblycostdominantcase,balanceddisassemblyand recoverycostscase,andrecoverycostdominantcased,b,r.Totalcostofthe disassemblyandrecoveryoperations c d + c r isxedatlowLandhighHlevels .5%and15%.Thetotalcostisthenallocatedtotheoperationssuchthatifthecosts areimbalanced,thenthecostofthedominantoperationissettwicethecostoftheother operation.Underthebalancedscheme,thetotalcostisallocatedtooperationsequally. Toprovidearichersetting,weconsidernormalanduniformdistributionsforthe demand.Basedontheresultsofourexperiments,themeanofthedemanddistribution, ,affectsthemagnitudeoftheprots,acquisitionquantities,andallocationquantities proportionaltoitsvalue.Thevarianceofthedemandparameter, 2 ,reducesthesystem protsandincreasestheacquisitionquantitytooffsettheincreaseddemandvariability 43

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asexpected.Therefore,wex atasinglevalueunitsperseason,and at10%of themeandemandwhentheunderlyingdistributionisnormal. Weconsiderafactorialdesignsettingforthediscountsusedtocompensatethe waitingtimeofacustomerforthenecessaryoperationswhenthedemandcannotbe satisedimmediately.Forthispurpose,wesetthelargerdiscount, e ,attwolevels %and10%.Thesmallerdiscount, d ,isvariedat25%and50%of e .Theunit loss-of-goodwillcostisxedat20%.Theunitsalvagevaluesaresetat3%forthe reusablepartand10%forthehulk. Inournumericalstudy,foragivensellingprice-demanddistributionpairandunder stochasticdemanufacturingyield,wehave11,520probleminstanceswhenthesupply ofend-of-lifeproductsisacquisitionpricedependent,and5,760probleminstances otherwise.Similarly,foragivenprice-demanddistributionpairandunderdeterministic demanufacturingyield,wehave2,304probleminstanceswhenthesupplyofend-of-life productsisacquisitionpricedependent,and1,152probleminstancesotherwise.Since wehaveasinglethree-stockandthreesingle-stock-pointmodels,wehaveatotalof 331,776probleminstances.Table3-6summarizesourparameterset: Table3-6.Parametersetforthesingle-periodnewsvendormodels. pabcc i Y CV Y h p Intensity c d + c r edr v v h 501020300.10.700.00610 d 7.5301052520 ,10 200300.50.750.01221.5 b 151050 0.800.01823 r 0.850.02436 0.900.0304 0.95 %of: apppc d + c r p pepp Usinganumericalstudywewereabletoquantifyouranalyticalndings.From alloftheexperimentsconducted,weidentiedhowallmodelsperformandranked themunderdifferentparametersettingsseeTables3-7and3-8.Notethatthe three-stock-pointmodelisthebestmodel,andhenceitisomittedinthesetables.The starsymbol meansthattheleadingmodelstartsasthesecondbestmodelafterthe 44

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three-point-stockmodelatlowdemanufacturingyieldrates,andperformsworsethan thefollowingmodelathigherdemanufacturingyieldrates.Weshowsingle-stock-point models 1S-R 1S-D ,and 1S-P with R D ,and P toshortenthetables. Table3-7.Rankingsofsingle-stock-pointmodelsunderacquisition-price-independent supply. NoenvironmentalfeesPositiveenvironmentalfees % c d + c r :LowHighLowHigh e d dbrdbrdbrdbr L L DRPDRPDRPPDRP DRDPRDRPDRPDRPDRPDRPDRP L H RPDRDPDRPPRDPDRP*D*RRDPRDPDRPRDPDRPDRP H L RDPRDPDRPRDPDRPDRPRDPRDPDRPRDPDRPDRP H H RDPRDPRDPRD*P*RDPD*R*PRDPRDPRDPRDPRDPR DP Table3-8.Rankingsofsingle-stock-pointmodelsunderacquisition-price-dependent supply. NoenvironmentalfeesPositiveenvironmentalfees % c d + c r :LowHighLowHigh e d dbrdbrdbrdbr L L RDPRDPR DPR DPDRPDRPRDPRDPR DPR DPDRPDRP L H RDPRDPRDPRD PRDPR DPRDPRDPRDPRDPRDPR DP H L RDPRDPRDPRDPRDPR DPRDPRDPRDPRDPRDPR DP H H RDPRDPRDPRDPRDPRDPRDPRDPRDPRDPRDPRDP Table3-9summarizestheminimumandmaximumrelativeprotimprovementsi.e., therangeofthepercentageprotimprovementsobtainedby 3S comparedto 1S over allparametersettings.Notethatwesuppressthelettersinmodels'namesreferringto stochasticityofthedemanufacturingyieldandthetypeofsupplytoshortenthetables. Table3-9.Rangesofpercentageprotimprovementofthree-stock-pointmodelsover single-stock-pointmodels, p =50 =30 Acquisition-price-independentsupplyAcquisition-price-dependentsupply Yield:DeterministicStochasticDeterministicStochastic DemandModelminmaxminmaxminmaxminmax Uniform 1S-P 3.74134.693.75134.964.21169.764.21169.89 1S-D 0.4120.880.4120.900.4423.260.4423.27 1S-R 0.0016.640.0016.650.004.160.004.17 Normal 1S-P 8.1355.508.1155.718.02104.458.00104.45 1S-D 1.8212.421.8112.451.8020.081.7920.08 1S-R 0.001.200.001.260.000.240.000.25 Modelswithdeterministicdemanufacturingyieldrateprovideupperboundsonthe expectedprotsobtainedunderthestochasticdemanufacturingyieldrate,whereasthe 45

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relativeperformancesandthebehaviorsofthemodelsstaythesameunderbothcases. Furthermore,highercoefcientofvariationvaluesofthestochasticdemanufacturing yieldrateslightlylowerstheexpectedprots,increasestotalacquisitionquantity tocopewiththeincreaseduncertainty,andmayslightlydecreasetheacquisition prices.Therefore,modelswithdeterministicdemanufacturingyieldratecanbeused toapproximatethestochasticversionssinceexplicitsolutionsundersomecasesmay eitherbeavailableasintheacquisition-price-independentsupplycase,ormaybe analyticallymoretractable.Wefurtherobservedthatthediagnostictestcosthasno signicanteffectonthevaluesofthedecisionvariablesotherthanslightlyreducingthe totalprots. Itcanbeseenthatthedemanddistribution,inparticularthevarianceofthedemand distribution,affectstherelativeperformancesbutnottheoverallbehaviorsofthemodels. Undernormaldemand,allmodelsperformasifthedemandisdeterministicduetolow varianceselectedinordertopreventnegativedemandcases.Therangesofpercentage protdiscrepanciesdecreaseforallsingle-stock-pointmodels,wheretheminimum relativediscrepanciesincreaseforallbut 1S-R ,andthemaximumrelativediscrepancies decreaseundernormaldemanddistribution.Intheremainderofthisdissertation,we willfocusontheresultsoftheexperimentsinwhichthedemandisuniformlydistributed sincetheoverallsystembehaviorstaysthesameunderbothdemanddistributionsand sincethisgivesusbetteropportunitytocapturetheeffectsofthedemanduncertainty. FromTable3-9,itisalsoclearthatthethree-stock-pointmodeloutperformsany othersingle-stock-pointmodeleitherintheformofstockingonlytheend-of-lifeproducts orthereusablepartsdisassembledorrecovered.Inotherwords,postponingoneor bothdemanufacturingoperationsisinferiortoprioritizingthem,especiallyunderpositive environmentalfeesandhighlevelsofoperationalcostsandapplicablediscounts.The protgainsfromusingathree-stock-pointmodelascomparedtothetraditionalpractice ofstockingend-of-lifeproductsexceed100%inseveralcases.Stockingonlyrecovered 46

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partsappearsasaviablepractice,yieldingprotsclosetothatofthethree-stock-point modelwithadiscrepancyofatmost4.17%underacquisition-price-dependentand 16.65%underacquisition-price-independentsupply. Figure3-2displaystherelativeprotimprovementsof S3S oversingle-stock-point modelsundervariouslevelsofdiscountsandtheabsenceofenvironmentalfees: Figure3-2.Relativeprotimprovementof S3S oversingle-stock-pointmodels, X U,60, p =50 c d + c r =15% c i =0.1% CV Y =0.0304 h p =0 Intheabsenceofenvironmentalfees, 3S and 1S-P haveonelesscostitemto dealwith,whereasthishasnoeffectontheothersingle-stock-pointmodels.Despite 47

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theiradvantageofnotbeingsubjecttotheenvironmentalfees,othersingle-stock-point models,i.e., 1S-D and 1S-R ,maysufferfromtheoperationalcostsandtheirweights, formeralsofromtheapplicablediscount.Inotherwords,stockingandoperational structuresofthesingle-stock-pointmodelsdeterminethecoststhesemodelsare subjectto,andaccordinglydeterminetheirperformances. Figure3-3displaystherelativeprotimprovementsof S3S oversingle-stock-point modelsundervariouslevelsofdiscountsandpositiveenvironmentalfees: Figure3-3.Relativeprotimprovementof S3S oversingle-stock-pointmodels, X U,60, p =50 c d + c r =15% c i =0.1% CV Y =0.0304 h p =3% 48

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Inthepresenceofenvironmentalfees, 1S-P performssignicantlyworse.Itisthe theworstoverallmodelunderallparametersettingsasitissubjecttothisadditional cost,anddoesnothavetheexibilityofthethree-stock-pointmodeltoallocatethe acquireditemstomultiplestockpointstodispersethecostsefciently.Therefore, postponingbothoperationsisclearlyinferiortoprioritizingoneorbothoperationsunder thepresenceofenvironmentalfees.Inotherwords,thetraditionalpracticeofpilingup end-of-lifeproductsisseverelyandadverselyaffectedfromthetrade-offbetweenthe environmentalfeesandtheoperationalcosts. 49

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CHAPTER4 ASSESSINGTHEVALUEOFDIAGNOSTICTESTINGANDYIELDINFORMATIONIN DEMANUFACTURING 4.1BackgroundandMotivation Ademanufacturer,whoacquiresend-of-lifeproductstoreclaimreusablepartsor componentsonsuchproductsforthepurposeofrecoveringandthensellingthemin aused-partsmarket,oftenfacesuncertaintyabouttherecoverabilityoftheseparts orcomponents.Demanufacturingoperationssuchasdisassemblyoftheend-of-life producttoextractreusableparts,andrecoveryofthesepartsareusuallyexpensivedue tobeinglaborintensive.Therefore,informationabouttheconditionofareusableparton anend-of-lifeproductmayhelpthedemanufactureravoidtheseexpensiveoperationsin casethepartcanbecorrectlydiagnosed. Informationabouttheconditionofpartsonend-of-lifeproductsmaysometimesbe obtainedatafeefromtheenduserorbyconductingadiagnostictest.Forinstance,an end-of-lifevehiclesalvager,whoacquiresaparticularvehicletorebuildthetransmission onit,mayuseanelectronicdiagnostictoolthatreadstheconditionofthetransmission fromanon-boardcomputerinstalledinthevehicle.Thedemanufacturermayavoidthe expensivecostofproductdisassemblyifthetransmissionisidentiedasunrecoverable. However,suchadiagnostictestmaynotalwaysbeaccurate,andmaybesubject toclassicationerrors.Inotherwords,thediagnostictestmayyieldafalsepositiveor afalsenegativeconclusioni.e.,onethatfalselyidentiesapartasrecoverable,and viceversa.Similarorotherformsofdiagnostictests,whichmaybeconductedafterthe productisdisassembledandthepartisextracted,maystillyieldfalseconclusions.In fact,thetrueconditionofapartmaynotberevealeduntilaftertherecoveryoperation iscompleted.Asaresult,someend-of-lifeproductswithactuallyrecoverablepartson themmaybesalvaged,whereassomeotherswithactuallyunrecoverablepartsmaybe keptinstockuntiltheirrecoverabilityisrevealed,bothcasesresultinginunnecessary costs. 50

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Inadditiontothecostsofdemanufacturingoperations,thedemanufacturermay alsohavetopaythegovernmentenvironmentalfeesforstoringend-of-lifeproductsthat containhazardousmaterialsand/orcomponentse.g.,sulphurandleadinlead-acid batteries,engineoil,transmissionuid,etc.,inend-of-lifevehicles.Regulationof storingsuchend-of-lifeproductsbyimposingenvironmentalfeesmaymotivatethe demanufacturertoprioritizedemanufacturingoperationstoextractthereusablepart andrecoveritimmediatelyuponacquisitionandtosalvagethehulkasopposedto storingtheend-of-lifeproduct.However,ifdemanufacturingoperationsaretoocostlyto perform,thedemanufacturermayprefertopostponethemuntilademandoccurs. Inthiscontext,theaccuracyandstagingofthediagnostictestmaybecomemore importantinordertoefcientlyrecovertheenvironmentalandeconomicalvalues residinginend-of-lifeproducts.Ahighlyaccuratediagnostictestmaybeconducted uponacquisitionofend-of-lifeproductsandpriortoallocationofitemstostockpoints inordertoeliminatesomeoftheenvironmentalfeesanddemanufacturingcostsby salvagingtheproductsidentiedwithrecoverableandunrecoverableparts.However,if thediagnostictestisnotveryaccurate,thenthedemanufacturermayprefertopostpone ituntilthepartisextractedornottoconductitatallinordertopreventlossdueto classicationerrors.Thedemanufacturermayhavetotakeintoaccounttheaccuracy andcostofthediagnostictest,aswellasthedemanufacturingyieldrateindeciding whethertoconductthetest,anditsstagingifitgoingtobeperformed. Inthispartofthedissertation,weinvestigatethevalueofthediagnostictestingand thedemanufacturingyieldrateinformationontheprotabilityofthedemanufacturing business.Tothisend,weutilizethetheoreticalndingsofourrstresearch,i.e.,we usethethree-stock-pointstructureasthebasicsetting.Ourgoalistoinvestigateand comparediagnostictestingschemesthatdifferbytheirstagingwitheachotheraswell aswithano-testingscenariousingrealisticmathematicalmodelsinordertoassessthe valueofthediagnostictestingindemanufacturing. 51

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4.2ProblemDenition Similartoourrstresearchproblem,weconsiderademanufacturerwhoacquires aparticularend-of-lifeproductfromtheend-usermarketinordertorecoveraparticular reusablepartonittosellintheused-partsmarket.Arecoveredpartisonethatwent throughbothdemanufacturingoperationsi.e.,disassemblyandrecovery,andis suitableforsaleintheused-partsmarket. Thedemanufacturermayconductadiagnostictestuponacquisitionorafterproduct disassemblytoidentifytherecoverabilityofthereusablepartontheend-of-lifeproduct; ormaynotconductadiagnostictest.Thediagnostictestmaynotbeperfect,i.e.,may haveclassicationerrors,andmayresultinfalseconclusions.Inthiscase,theallocated itemsmaycontainfalsepositivesand/orfalsenegatives.Thetruerecoverabilityof areusablepartisrevealedupontherecoveryoperation.Theprobabilityofcorrectly identifyingatrulyrecoverableitemisdenotedby 2 ,1] ,whereascorrectlyidentifying atrulyunrecoverableitemisdenotedby 2 ,1] .Both and areassumedtobe knownwithcertainty. Acquiredend-of-lifeproductsmaybeallocatedamongthreestockpointsafter thenecessarydemanufacturingoperationsarecompleted.Allocationquantitiesof end-of-lifeproducts,disassembledparts,andtherecoveredpartscanbederived dependingonthedemanufacturingyieldrate y 2 ,1 ,accuracyandthestageof thediagnostictest.Figure4-1showstheresultsofthetest,i.e.,thequantityofitems decidedtobekeptandthosetobesalvaged: Figure4-1.Quantityoftheacquiredend-of-lifeproductsallocatedandsalvagedafter diagnostictestingisconducted. 52

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Weconsidertwodiagnostictestingschemes,andabenchmarkcaseinwhichno diagnostictestisconductedseeFigure4-2,wherewelabeltheacquisitionquantity ofend-of-lifeproductstoallocateamongthestockpointsforend-of-lifeproductsas P ,disassembledpartsas D ,andrecoveredpartsas R .In preliminarytesting ,the diagnostictestisconductedimmediatelyafteracquisitionbutpriortoallocation.In postponedtesting ,thediagnostictestisconductedafterthedisassemblyoperation. A B C Figure4-2.Flowofitemsinthesystemundertestingandnotestingscenarios. APreliminarytesting.BPostponedtesting.CNotesting. 53

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Inthe preliminarytesting scenarioseeFigure4-2A,theacquisitionpriceis determined,end-of-lifeproductsareacquired,andadiagnostictestisconductedto assesswhetherthereusablepartsonthemarerecoverableatacertaintestingcost. Thisisthestochasticthree-stock-pointmodelconsideredinourrstresearchproblem, exceptthatwenowconsiderthepossibilityoffalseconclusionsofthediagnostictest. Withanimperfectdiagnostictest,someend-of-lifeproductswithactuallyrecoverable partsonthemmaybesalvaged,whereassomeotherend-of-lifeproductswithactually unrecoverablepartsonthemmaybeallocatedtostockpointsofend-of-lifeproductsand disassembledbutnotrecoveredparts.Sincetruerecoverabilityofapartisrevealedwith certaintyupontherecoveryoperation,thosepartsthatareallocatedtothestockpointof therecoveredpartsareappropriateforsale. Afterthediagnostictestisconducted,thoseend-of-lifeproductsthatarediagnosed asunrecoverablecanbesoldtoamaterialrecycler.Ifthetestisimperfect,these productsthataresoldfortheirmaterialcontentsmayincludefalsenegatives.Similarly, thosethatpasstheimperfectdiagnostictestmaycontainfalsepositives,andare allocatedamongthethreestockpointsinthesystemasdescribedbelow: 1.Someofthemthatpassthediagnostictestareallocatetothestockpointfor end-of-lifeproducts,costingthefacilitytheassociatedenvironmentalfees.These end-of-lifeproductsmaycontainfalsepositivesifthediagnostictestisimperfect. 2.Someofthem,whichpassthetest,mayundergoonlythedisassemblyoperation toremovethereusableparts,andthefacilityincurstheassociatedcostsof thedisassemblyoperation.Theextractedpartsarethenallocatedtothestock pointfordisassembledparts.Similartoallocatedend-of-lifeproducts,stockof disassembledpartsmaycontainfalsepositivesifthediagnostictestisimperfect. Thehulkwhichmaycontainfalsenegativesincaseofanimperfectdiagnostic testcanbesoldtoamaterialrecyclerforitsmaterialcontent. 3.Therestmaygounderboththedisassemblyandtherecoveryoperations.After thepartsareextracted,theygointotherecoveryoperationtoberestoredfor reuse.Incaseofanimperfectdiagnostictest,truerecoverabilityofapartis revealedupontherecoveryoperation.Ifthepartcannotberecovered,thenitcan besoldtothematerialrecycler.Otherwise,thepartisallocatedtothestockpoint forrecoveredparts.Thehulkcanagainbesoldtothematerialrecycler. 54

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Inthe postponedtesting scenarioseeFigure4-2B,testforrecoverabilityis conducteduponthecompletionofthedisassemblyoperation.Inthisscenario,rst, acquisitionpriceisdeterminedandend-of-lifeproductsareacquired,andthen,these productsareallocatedamongthethreestockpointsinthesystemasfollows: 1.Someofthemarekeptintheend-of-lifeproductsstock,costingthefacilitythe associatedenvironmentalfees. 2.Someofthemmayundergoonlythedisassemblyoperationtoremovethe reusableparts,andthefacilityincurstheassociatedcostsofthedisassembly operation.Theseextractedpartsarethentestedatacertaincost,andthosethat passthetestareallocatedtothestockpointfordisassembledparts,whereasthe hulkcanbesoldtothematerialrecyclerforitsmaterialcontent.Similartothe preliminarytestingscenario,theallocatedpartsmaycontainfalsepositivesand thosesalvageditemsmaycontainfalsenegativesifthetestisimperfect. 3.Therestmaygounderboththedisassemblyandtherecoveryoperations.After thepartsareextracted,theyaretested,andthetestingcostsareincurred.Those partsthatpassthetestfurthergointotherecoveryoperationtoberestoredfor reuse.Sincetherecoveryoperationidentiesthetruerecoverabilityoftheparts, thosepartsthatcannotberecoveredaresoldtothematerialrecycler.Recovered partsarestored,andthehulkcanagainbesoldtothematerialrecycler. Inthebenchmarkscenario,nodiagnostictestisconductedseeFigure4-2C,and hence,notestingcostisincurred.Truerecoverabilityisrevealedaftertherecovery operation.Acquiredend-of-lifeproductsareallocatedinthesystemasfollows: 1.Someofthemarekeptintheend-of-lifeproductsstock,costingthefacilitythe associatedenvironmentalfees. 2.Someofthemmayundergoonlythedisassemblyoperationtoremovethe reusableparts,andthefacilityincurstheassociatedcostsofthedisassembly operation.Theseextractedpartsarethenallocatedtothestockpointfor disassembledparts,whereasthehulkcanbesoldtothematerialrecycler. 3.Therestmaygounderboththedisassemblyandtherecoveryoperations.The facilityincursthecostsofbothoperations.Ifthepartcannotberecovered,itcan besoldtothematerialrecycler.Otherwise,thepartisallocatedtothestockpoint forrecoveredparts.Thehulkcanagainbesoldtothematerialrecycler. Ourdecisionvariablesaresimilartothoseinourrstresearch:theunitacquisition cost C tosetonend-of-lifeproducts;andthequantitiesofitemstoallocateacrossthe stocksof:end-of-lifeproducts P ,disassembledparts D ,andrecoveredparts R 55

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4.3ModelDevelopmentandAnalysis Welabel preliminarytestingmodel as PTS postponedtestingmodel as DTS and notestingmodel as NTS ;indicatingthatthediagnostictestisconductedon end-of-lifeproductsuponreceipt,extractedpartsupondisassembly,andisnot conducted,respectively.Let y = y + )]TJ/F25 11.9552 Tf 13.138 0 Td [( )]TJ/F39 11.9552 Tf 13.138 0 Td [(y anditsexpected value y = E [ Y ]+ )]TJ/F25 11.9552 Tf 12.866 0 Td [( )]TJ/F39 11.9552 Tf 12.866 0 Td [(E [ Y ] forfuturederivations. y is thepercentageoftheacquiredquantityofend-of-lifeproductsallocatedamongstock pointsuponthediagnostictest.Thersttermcorrespondstotheproportionoftrue positives,andthesecondtermistheproportionoffalsepositivesseeFigure4-1for derivationsoftheallocationquantities.Noticethatthispercentageshouldbepositive, requiring + 1 ,i.e.,correctlyidentifyingtruepositivesandnegativestoexceed theunity.Thisrequirementisameaningfulassumptionfrequentlyusedinliterature seeZikopoulosandTagaras2008,whichstatesthatanitemidentiedasrecoverable ismorelikelytoberecoverablethanonethatisidentiedasunrecoverable.Table4-1 liststheallocationquantitiesofacquiredend-of-lifeproducts P D R amongstock pointsunderallscenarios,wherewesuppresstheargumentsof forsimplicity: Table4-1.Allocationquantitiesofacquiredend-of-lifeproductsamongstockpoints. ModelEnd-of-lifeproductsDisassembledpartsRecoveredparts PreliminaryTesting P D yR PostponedTesting P D yR NoTesting PDyR Wealsoconsiderspecialcaseswithacquisition-price-independentsupply, andweadd IS tothefrontofthemodel'slabelstoindicatethis.Thesemodelsare straightforwardtoobtainbyreplacingtheacquisitionpricetermintheprotfunctionof themodelsunderacquisition-price-dependentsupplywiththeparameter, c .Modelsin whichthedemanufacturingyieldrateisdeterministichave D attheendofmodels'labels insteadof S .Similartoourrstresearchproblem,werstpresentourformulations underthestochastic,andthenunderthedeterministicdemanufacturingyieldrate. 56

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4.3.1StochasticYieldRate Inthissection,wepresentmodelsforthestochasticdemanufacturingyieldrate whenthesupplyofend-of-lifeproductsdependsontheacquisitionprice.Expected protsforallmodelscanbederivedusingtheallocationquantitiesgiveninTable4-1, stageofthediagnostictestifitisconducted,andtheidentitybetweenthetotal acquisitionquantity R + D + P andthesupplyofend-of-lifeproductsthatcanbe acquiredbythethedemanufacturer a + bC .Theprotfunctionofthepreliminary testingmodelissimilartothatoftheperfectpreliminarytestingmodelwherethe quantitiesoftrulyrecoverableitemsaremodiedusingtheaccuracyofthediagnostic test.Equations4-1,4-2,and4-3presentmodels'expectedtotalprotfunctions: PTS : PTS R D P = Z y U y L Z yR 0 pxf x dx + Z y R + D yR )]TJ/F39 11.9552 Tf 5.48 -9.683 Td [(p yR + p )]TJ/F39 11.9552 Tf 11.956 0 Td [(d )]TJ/F39 11.9552 Tf 11.956 0 Td [(c r = y x )]TJ/F25 11.9552 Tf 11.955 0 Td [( yR f x dx + Z y R + D + P y R + D p yR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r = y yD + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F22 11.9552 Tf 11.956 0 Td [( c d + c r = y x )]TJ/F25 11.9552 Tf 11.955 0 Td [( y R + D f x dx + Z 1 y R + D + P y )]TJ/F39 11.9552 Tf 5.479 -9.683 Td [(pR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r = y D + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F22 11.9552 Tf 11.955 0 Td [( c d + c r = y P f x dx + Z y R + D + P 0 v )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( y R + D + P )]TJ/F39 11.9552 Tf 11.955 0 Td [(x f x dx )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z 1 y R + D + P r )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(x )]TJ/F25 11.9552 Tf 11.955 0 Td [( y R + D + P f x dx )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p P )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d R + D )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r R + v )]TJ/F25 11.9552 Tf 11.955 0 Td [( y R + D + P g y dy )]TJ/F22 11.9552 Tf 11.955 0 Td [( R + D + P )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b R + D + P )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i R + D + P + v h R + D + P -1 DTS : DTS R D P = Z y U y L Z yR 0 pxf x dx + Z y R + D yR )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(p yR + p )]TJ/F39 11.9552 Tf 11.956 0 Td [(d )]TJ/F39 11.9552 Tf 11.956 0 Td [(c r = y x )]TJ/F25 11.9552 Tf 11.955 0 Td [( yR f x dx + Z y R + D + P y R + D p yR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r = y yD 57

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+ p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F22 11.9552 Tf 11.955 0 Td [( c d + c i = y )]TJ/F39 11.9552 Tf 11.956 0 Td [(c r = y x )]TJ/F25 11.9552 Tf 11.955 0 Td [( y R + D f x dx + Z 1 y R + D + P y )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(pR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r = y D + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F22 11.9552 Tf 11.956 0 Td [( c d + c i = y )]TJ/F25 11.9552 Tf 11.955 0 Td [( c r = y P f x dx + Z y R + D + P 0 v )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [( y R + D + P )]TJ/F39 11.9552 Tf 11.955 0 Td [(x f x dx )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z 1 y R + D + P r )]TJ/F39 11.9552 Tf 5.479 -9.683 Td [(x )]TJ/F25 11.9552 Tf 11.955 0 Td [( y R + D + P f x dx )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r R + v )]TJ/F25 11.9552 Tf 11.955 0 Td [( y R + D + P g y dy )]TJ/F22 11.9552 Tf 11.955 0 Td [( R + D + P )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b R + D + P )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i R + D )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p P )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d R + D + v h R + D + P -2 NTS : NTS R D P = Z y U y L Z yR 0 pxf x dx + Z y R + D yR )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(pyR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r = y x )]TJ/F39 11.9552 Tf 11.955 0 Td [(yR f x dx + Z y R + D + P y R + D )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(pyR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r = y yD + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F22 11.9552 Tf 11.955 0 Td [( c d + c r = y x )]TJ/F39 11.9552 Tf 11.955 0 Td [(y R + D f x dx + Z 1 y R + D + P y )]TJ/F39 11.9552 Tf 5.48 -9.683 Td [(pR + p )]TJ/F39 11.9552 Tf 11.956 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r = y D + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F22 11.9552 Tf 11.955 0 Td [( c d + c r = y P f x dx + Z y R + D + P 0 v )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(y R + D + P )]TJ/F39 11.9552 Tf 11.955 0 Td [(x f x dx )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z 1 y R + D + P r )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(x )]TJ/F39 11.9552 Tf 11.955 0 Td [(y R + D + P f x dx + v )]TJ/F39 11.9552 Tf 11.955 0 Td [(y R + D + P g y dy )]TJ/F22 11.9552 Tf 11.955 0 Td [( R + D + P )]TJ/F39 11.9552 Tf 11.956 0 Td [(a = b R + D + P )]TJ/F39 11.9552 Tf 11.956 0 Td [(h p P )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d R + D )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r R + v h R + D + P -3 Theexpectedprotfunctionsinequations4-1,4-2,and4-3arestrictlyconcave intheacquisitionquantitiesasstatedinTheorems4.1,4.2,and4.3below: Theorem4.1. PTS R D P isstrictlyconcavein R D and P ifandonlyifthe followingconditionsholdforall y 2 y L y U :i y p )]TJ/F39 11.9552 Tf 10.272 0 Td [(e )]TJ/F30 11.9552 Tf 10.271 9.684 Td [()]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( y + )]TJ/F25 11.9552 Tf 10.272 0 Td [( )]TJ/F39 11.9552 Tf 10.272 0 Td [(y c d + c r > y v )]TJ/F39 11.9552 Tf 11.995 0 Td [(r ,i.e.,satisfyingthereusablepartdemandfromthestockofend-of-lifeproducts ismoreprotablethansalvagingthepart,earningthesalvagerevenueandincurringthe penaltyforthelostsale,whichleadsto Y > y LB PTS = )]TJ/F25 11.9552 Tf 10.129 0 Td [( c d + c r = )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( p )]TJ/F39 11.9552 Tf 10.13 0 Td [(e )]TJ/F39 11.9552 Tf 10.129 0 Td [(c d )]TJ/F39 11.9552 Tf 10.13 0 Td [(c r )]TJ/F39 11.9552 Tf 10.13 0 Td [(v + r + )]TJ/F25 11.9552 Tf 10.66 0 Td [( c d + c r ,requiring p )]TJ/F39 11.9552 Tf 10.66 0 Td [(e )]TJ/F39 11.9552 Tf 10.659 0 Td [(c d )]TJ/F39 11.9552 Tf 10.659 0 Td [(c r > v )]TJ/F39 11.9552 Tf 10.66 0 Td [(r ,and y L > y LB PTS ,i.e.,thelowerbound onthedemanufacturingyieldratetoexceedacriticalpercentagegivenbytheproportion oftheoperationalcostsincurredforafalsepositiveduetoincorrectlyidentifyingitas recoverabletothetotalcostincurredforacorrectlyidentiedtruepositiveinadditionto 58

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theformercost,ii e + c d + c r = y > d + c r = y ,i.e.,themarginalcostofpostponing thedisassemblyoperationisgreaterthanthemarginalcostofpostponingtherecovery operation,andiii d + c r = y > 0 ,i.e.,themarginalcostofpostponingtherecovery operationispositive. Proof. Firstorderoptimalityconditionsaregivenbelow: PTS 0 R = )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [( y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( c d + c r F y R + D + P g y dy )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [( y e )]TJ/F39 11.9552 Tf 11.956 0 Td [(d + c d F y R + D g y dy )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z y U y L yd + c r F yR g y dy )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = b + E [ Y ] p )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F22 11.9552 Tf 11.955 0 Td [( c d + c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h + a = b PTS 0 D = )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( c d + c r F y R + D + P g y dy )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y e )]TJ/F39 11.9552 Tf 11.955 0 Td [(d + c d F y R + D g y dy )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = b + E [ Y ] p )]TJ/F39 11.9552 Tf 11.956 0 Td [(d )]TJ/F39 11.9552 Tf 11.956 0 Td [(v + r )]TJ/F22 11.9552 Tf 11.955 0 Td [( c d + c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h + a = b PTS 0 P = )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( c d + c r F y R + D + P g y dy )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = b + E [ Y ] p )]TJ/F39 11.9552 Tf 11.956 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F22 11.9552 Tf 11.955 0 Td [( c d + c r + h p )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h + a = b TheHessianmatrix H hasthesameformgivenintheproofofTheorem3.1with A = R y U y L y )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( y p )]TJ/F39 11.9552 Tf 12.663 0 Td [(e )]TJ/F39 11.9552 Tf 12.663 0 Td [(v + r )]TJ/F25 11.9552 Tf 12.663 0 Td [( c d + c r f y R + D + P g y dy +2 = b B = R y U y L y )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y e )]TJ/F39 11.9552 Tf 11.248 0 Td [(d + c d f y R + D g y dy ,and C = R y U y L y )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( y + c r f yR g y dy H isnegativedeniteifandonlyifthethetermsinparenthesesinsidetheintegralsfor A B C arepositive,fromwhichtherequiredconditionsforconcavityareobtainedgiven that + 1 with 2 ,1] Wefurthernotethatconditionsiiandiiiareunnecessaryunderthenon-trivial coststructureandtheaccuracyconstraintsofthediagnostictest;andaremerely presentedforthesakeofcompleteness. 59

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Theorem4.2. DTS R D P isstrictlyconcavein R D and P ifandonlyifthe followingconditionsholdforall y 2 y L y U :i y p )]TJ/F39 11.9552 Tf 9.541 0 Td [(e )]TJ/F39 11.9552 Tf 9.541 0 Td [(c d )]TJ/F39 11.9552 Tf 9.541 0 Td [(c i )]TJ/F30 11.9552 Tf 9.541 9.684 Td [()]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y + )]TJ/F25 11.9552 Tf 9.54 0 Td [( )]TJ/F39 11.9552 Tf 9.54 0 Td [(y c r > y v )]TJ/F39 11.9552 Tf 11.994 0 Td [(r ,i.e.,satisfyingthereusablepartdemandfromthestockofend-of-lifeproducts ismoreprotablethansalvagingthepart,earningthesalvagerevenueandincurringthe penaltyforthelostsale,whichleadsto Y > y LB DTS = )]TJ/F39 11.9552 Tf 5.479 -9.683 Td [(c d + c i + )]TJ/F25 11.9552 Tf 11.796 0 Td [( c r = )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( p )]TJ/F39 11.9552 Tf 11.796 0 Td [(e )]TJ/F39 11.9552 Tf 11.796 0 Td [(c r )]TJ/F39 11.9552 Tf -458.701 -23.907 Td [(v + r + )]TJ/F25 11.9552 Tf 12.044 0 Td [( c r ,requiring p )]TJ/F39 11.9552 Tf 12.044 0 Td [(e )]TJ/F39 11.9552 Tf 12.044 0 Td [(c r )]TJ/F39 11.9552 Tf 12.044 0 Td [(c d )]TJ/F39 11.9552 Tf 12.044 0 Td [(c i > v )]TJ/F39 11.9552 Tf 12.044 0 Td [(r ,and y L > y LB DTS ,i.e.,the lowerboundonthedemanufacturingyieldratetoexceedacriticalpercentagegivenby theproportionoftheinspectionandoperationalcostsincurredforafalsepositivedueto incorrectlyidentifyingitasrecoverabletothetotalcostincurredforacorrectlyidentied truepositiveinadditiontotheformercost,ii e + c d + c i = y + c r = y > d + c r = y ,i.e., themarginalcostofpostponingthedisassemblyoperationisgreaterthanthemarginal costofpostponingtherecoveryoperation,andiii d + c r = y > 0 ,i.e.,themarginalcost ofpostponingtherecoveryoperationispositive. Proof. Firstorderoptimalityconditionsaregivenbelow: DTS 0 R = )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i )]TJ/F25 11.9552 Tf 11.955 0 Td [( c r F y R + D + P g y dy )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y e )]TJ/F39 11.9552 Tf 11.955 0 Td [(d + c d + c i F y R + D g y dy )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z y U y L yd + c r F yR g y dy )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = b + E [ Y ] p )]TJ/F39 11.9552 Tf 11.956 0 Td [(v + r )]TJ/F22 11.9552 Tf 11.955 0 Td [( c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h + a = b DTS 0 D = )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.956 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i )]TJ/F25 11.9552 Tf 11.955 0 Td [( c r F y R + D + P g y dy )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y e )]TJ/F39 11.9552 Tf 11.955 0 Td [(d + c d + c i F y R + D g y dy )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = b + E [ Y ] p )]TJ/F39 11.9552 Tf 11.956 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F22 11.9552 Tf 11.955 0 Td [( c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h + a = b DTS 0 P = )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [( y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.956 0 Td [(v + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( c d + c r F y R + D + P g y dy )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = b + E [ Y ] p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F22 11.9552 Tf 11.955 0 Td [( c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h + a = b 60

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TheHessianmatrix H hasthesameformgivenintheproofofTheorem3.1with A = R y U y L y )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y p )]TJ/F39 11.9552 Tf 12.416 0 Td [(e )]TJ/F39 11.9552 Tf 12.416 0 Td [(v + r )]TJ/F39 11.9552 Tf 12.416 0 Td [(c d )]TJ/F39 11.9552 Tf 12.416 0 Td [(c i )]TJ/F25 11.9552 Tf 12.416 0 Td [( c r f y R + D + P g y dy +2 = b B = R y U y L y )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( y e )]TJ/F39 11.9552 Tf 9.631 0 Td [(d + c d + c i f y R + D g y dy ,and C = R y U y L y )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y + c r f yR g y dy H isnegativedeniteifandonlyifthethetermsinparenthesesinsidetheintegralsfor A B C arepositive,fromwhichtherequiredconditionsforconcavityareobtainedgiven that + 1 with 2 ,1] .SimilartotheproofofTheorem4.2,conditionsiiand iiiarenotnecessary,andaremerelypresentedforthesakeofcompleteness. Theorem4.3. NTS R D P isstrictlyconcavein R D and P ifandonlyifthe followingconditionsholdforall y 2 y L y U :i y p )]TJ/F39 11.9552 Tf 12.703 0 Td [(e )]TJ/F39 11.9552 Tf 12.702 0 Td [(c d )]TJ/F39 11.9552 Tf 12.703 0 Td [(c r > y v )]TJ/F39 11.9552 Tf 12.702 0 Td [(r ,i.e., satisfyingthereusablepartdemandfromthestockofend-of-lifeproductsismore protablethansalvagingthepart,earningthesalvagerevenueandincurringthepenalty forthelostsalefor,whichleadsto Y > y LB NTS = c d + c r = p )]TJ/F39 11.9552 Tf 12.633 0 Td [(e )]TJ/F39 11.9552 Tf 12.633 0 Td [(v + r ,requiring p )]TJ/F39 11.9552 Tf 12.394 0 Td [(e )]TJ/F39 11.9552 Tf 12.394 0 Td [(c d )]TJ/F39 11.9552 Tf 12.395 0 Td [(c r > v )]TJ/F39 11.9552 Tf 12.394 0 Td [(r ,and y L > y LB NTS ,i.e.,thelowerboundonthedemanufacturing yieldratetoexceedacriticalpercentagegivenbytheproportionoftheoperationalcosts incurredforanunrecoverableparttothetotalcostincurredforarecoverablepartin additiontotheformercost,ii e + c d + c r = y > d + c r = y ,i.e.,themarginalcostof postponingthedisassemblyoperationisgreaterthanthemarginalcostofpostponing therecoveryoperation,andiii d + c r = y > 0 ,i.e.,themarginalcostofpostponingthe recoveryoperationispositive. Proof. Firstorderoptimalityconditionsaregivenbelow: NTS 0 R = )]TJ/F30 11.9552 Tf 11.956 16.273 Td [(Z y U y L )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r F y R + D + P g y dy )]TJ/F30 11.9552 Tf 11.956 16.273 Td [(Z y U y L )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(y e )]TJ/F39 11.9552 Tf 11.956 0 Td [(d + c d F y R + D g y dy )]TJ/F30 11.9552 Tf 11.956 16.273 Td [(Z y U y L yd + c r F yR g y dy )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = b + E [ Y ] p )]TJ/F39 11.9552 Tf 11.956 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r + v + v h + a = b NTS 0 D = )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z y U y L )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.956 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r F y R + D + P g y dy )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z y U y L )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(y e )]TJ/F39 11.9552 Tf 11.955 0 Td [(d + c d F y R + D g y dy 61

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)]TJ/F22 11.9552 Tf 11.956 0 Td [(2 R + D + P = b + E [ Y ] p )]TJ/F39 11.9552 Tf 11.956 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r + v + v h + a = b NTS 0 P = )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z y U y L )]TJ/F39 11.9552 Tf 5.479 -9.683 Td [(y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r F y R + D + P g y dy )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = b )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p + E [ Y ] p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r + v + v h + a = b TheHessianmatrix H hasthesameformgivenintheproofofTheorem3.1with A = R y U y L y )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(y p )]TJ/F39 11.9552 Tf 12.17 0 Td [(e )]TJ/F39 11.9552 Tf 12.171 0 Td [(v + r )]TJ/F39 11.9552 Tf 12.171 0 Td [(c d )]TJ/F39 11.9552 Tf 12.17 0 Td [(c r f y R + D + P g y dy +2 = b B = R y U y L y )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(y e )]TJ/F39 11.9552 Tf -451.114 -23.908 Td [(d + c d f y R + D g y dy ,and C = R y U y L y )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(y + c r f yR g y dy H isnegativedenite ifandonlyifthethetermsinparenthesesinsidetheintegralsfor A B C arepositive, fromwhichtherequiredconditionsforconcavityareobtained.Similartotheproofof Theorem4.2,conditionsiiandiiiaremerelygivenforcompleteness. Sincetheexpectedprotfunctionisstrictlyconcave,andissubjecttonon-negativity restrictions,KKTconditionsarebothnecessaryandsufcientforoptimality.Thesolution approachissimilartotheoneweemployedinourrstresearch.Inotherwords,ifthe stationarypointobtainedfromtherstorderconditionsisfeasible,thenitistheunique optimum,otherwise,afeasiblepointthatsatisestheKKTconditionsissearchedfor. KKTcasesarelistedinTables4-2,4-3,and4-4underageneraldemanddistribution: Table4-2.PTS:KKTcasesunderstochasticdemanufacturingyieldrateandgeneral demanddistribution. Case RDP PTS 0 R PTS 0 D PTS 0 P Equationstosolve C1 0 0 0 000 E [ Yd + c r F YR ]= E [ Y ] d E [ Y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d F Y R + D ] = E [ Y ] e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.963 0 Td [( c d + c r F Y R + D + P ] +2 R + D + P = b = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 7.748 0 Td [( c d + c r + h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h + a = b C4 0 0 0 0 < 0 0 E [ Ye + c d + c r F YR ]= E [ Y ] e + h p E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.963 0 Td [( c d + c r F Y R + P ] +2 R + P = b = E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 7.748 0 Td [( c d + c r + h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h + a = b C5 0 000 < 0 < 0 E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r F YR ]+2 R = b = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.962 0 Td [( c d + c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h + a = b C6 0 0 000 < 0 E [ Yd + c r F YR ]= E [ Y ] d E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.962 0 Td [( c r F Y R + D ] +2 R + D = b = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.963 0 Td [( c d + c r )]TJ/F39 9.9626 Tf 7.748 0 Td [(c i + v + v h + a = b 62

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Table4-3.DTS:KKTcasesunderstochasticdemanufacturingyieldrateandgeneral demanddistribution. CaseEquationstosolve C1 E [ Yd + c r F YR ]= E [ Y ] d E [ Y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d + c i F Y R + D ]= E [ Y ] e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i )]TJ/F11 9.9626 Tf 9.963 0 Td [( c r F Y R + D + P ]+2 R + D + P = b = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F22 9.9626 Tf 9.963 0 Td [( c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h + a = b C4 E [ Ye + c d + c i + c r F YR ]= E [ Y ] e + h p E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i )]TJ/F22 9.9626 Tf 9.963 0 Td [( c r F Y R + P ]+2 R + P = b = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F22 9.9626 Tf 9.963 0 Td [( c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h + a = b C5 E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r F YR ]+2 R = b = E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F22 9.9626 Tf 9.963 0 Td [( c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h + a = b C6 E [ Yd + c r F YR ]= E [ Y ] d E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.962 0 Td [( c r F Y R + D ]+2 R + D = b = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F22 9.9626 Tf 9.962 0 Td [( c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v h + v + a = b Table4-4.NTS:KKTcasesunderstochasticdemanufacturingyieldrateandgeneral demanddistribution. CaseEquationstosolve C1 E [ Yd + c r F YR ]= E [ Y ] d E [ Y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d F Y R + D ]= E [ Y ] e )]TJ/F39 9.9626 Tf 9.962 0 Td [(d + h p E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r F Y R + D + P ]+2 R + D + P = b = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r + v + v h + a = b C4 E [ Ye + c d + c r F YR ]= E [ Y ] e + h p E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c r F Y R + P ]+2 R + P = b = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r + v + v h + a = b C5 E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r F YR ]+2 R = b = E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r + v + v h + a = b C6 E [ Yd + c r F YR ]= E [ Y ] d E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c r F Y R + D ]+2 R + D = b = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r + v h + v + a = b Notethat y = y + )]TJ/F25 11.9552 Tf 12.225 0 Td [( )]TJ/F39 11.9552 Tf 12.226 0 Td [(y ,whereitsexpectedvalueisdenotedby ,andwesuppresstheirargumentstoshortenthetables.Furthermore,similartoour rstresearch,KKTcasesthatarenotsatisedbytheiranalyticalsolutionareomittedin tables. Availabilityofexplicitanalyticalsolutionsdependsonthecomplexityofthedemand distribution.Forcomplexdemanddistributionssuchasnormal,therstorder conditionsformasystemofnonlinearequations,whicharehardertosolve.Onthe otherhand,iftheformofthedemanddistributionissimplesuchasuniform,explicit analyticalsolutionsmayexist.KKTcasesandtheirsolutionsunderuniformlydistributed demandaresummarizedinTables4-5,4-6,and4-7forthemodelsweconsider: 63

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Table4-5.PTS:KKTcasesunderstochasticdemanufacturingyieldrateanduniform, U demand. CaseEquationstosolve C1 F R = E [ Y ] d = E [ Y 2 ] d + E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.963 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.962 0 Td [(E [ Y 2 ] c r F R + D = E [ Y ] e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p = E [ Y 2 ] e )]TJ/F39 9.9626 Tf 9.962 0 Td [(d + E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.963 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.962 0 Td [(E [ Y 2 ] c d F R + D + P ]= E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.963 0 Td [( c d + c r + h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.962 0 Td [( E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.962 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y 2 ] c d + c r +2 U = b C4 F R = E [ Y ] e + h p = E [ Y 2 ] e + E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.962 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y 2 ] c d + c r F R + P = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.963 0 Td [( c d + c r + h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.962 0 Td [( E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.962 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y 2 ] c d + c r +2 U = b C5 F R = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.963 0 Td [( c d + c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b = 2 E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r +2 U = b C6 F R = E [ Y ] d = E [ Y 2 ] d + E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.963 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.962 0 Td [(E [ Y 2 ] c r F R + D = E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.963 0 Td [( c d + c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h + a = b = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.962 0 Td [( E [ Y 2 ]+1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y 2 ] c r +2 U = b Table4-6.DTS:KKTcasesunderstochasticdemanufacturingyieldrateanduniform, U demand. CaseEquationstosolve C1 F R = E [ Y ] d = E [ Y 2 ] d + E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.963 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.962 0 Td [(E [ Y 2 ] c r F R + D = E [ Y ] e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p = E [ Y 2 ] e )]TJ/F39 9.9626 Tf 9.962 0 Td [(d + E [ Y ] c d + c i F R + D + P ]= E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.963 0 Td [( c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.962 0 Td [( E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.962 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y 2 ] c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y ] c d + c i +2 U = b C4 F R = E [ Y ] e + h p = E [ Y 2 ] e + E [ Y 2 ]+1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y 2 ] c r + E [ Y ] c d + c i F R + P = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F22 9.9626 Tf 9.963 0 Td [( c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.962 0 Td [( E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.962 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y 2 ] c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y ] c d + c i +2 U = b C5 F R = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F22 9.9626 Tf 9.962 0 Td [( c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b = 2 E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r +2 U = b C6 F R = E [ Y ] d = E [ Y 2 ] d + E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.963 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.962 0 Td [(E [ Y 2 ] c r F R + D = E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F22 9.9626 Tf 9.962 0 Td [( c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.962 0 Td [( E [ Y 2 ]+1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y 2 ] c r +2 U = b Table4-7.NTS:KKTcasesunderstochasticdemanufacturingyieldrateanduniform, U demand. CaseEquationstosolve C1 F R = E [ Y ] d = E [ Y 2 ] d + E [ Y ] c r F R + D = E [ Y ] e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p = E [ Y 2 ] e )]TJ/F39 9.9626 Tf 9.962 0 Td [(d + E [ Y ] c d F R + D + P ]= E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c r + v + v h + a = b = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(E [ Y ] c d + c r +2 U = b C4 F R = E [ Y ] e + h p = E [ Y 2 ] e + E [ Y ] c d + c r F R + P = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r + v + v h + a = b = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(E [ Y ] c d + c r +2 U = b C5 F R = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c r + v + v h + a = b = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r +2 U = b C6 F R = E [ Y ] d = E [ Y 2 ] d + E [ Y ] c r F R + D = E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c r + v + v h + a = b = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y ] c r +2 U = b KKTequationscanbeusedtoderivelowerboundsontheexpecteddemanufacturing yieldratethatarenecessaryforC1,C4,C5,andC6tohold.Theseboundsalongwith 64

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thoserequiredonthelowerboundofthedemanufacturingyieldrateforconcavityofthe expectedprotfunctionmayhelpthedemanufacturerobtainarangeofyieldratesover whichthebusinessisprotabletooperate,whichmayalsohelpthedemanufacturer selectacquisitionsourcesaccordingly.Wewillderivetheboundonlyfor PTS C1in Corollary4.3.1,andomitderivationsfortheothercasesandmodelssinceproofsare similar.AppendixDgiveslowerboundsforallmodelsandsomeexamples. Corollary4.3.1. Let = )]TJ/F22 11.9552 Tf 9.299 0 Td [( v + v h and = p )]TJ/F39 11.9552 Tf 12.604 0 Td [(v + r .Lowerboundson E [ Y ] forPTSinKKTcasesC1,C4,C5,andC6aregivenby:i,ii E [ Y ] max f 0, c i + + a = b + )]TJ/F25 11.9552 Tf 12.776 0 Td [( c d + c r + h p = )]TJ/F39 11.9552 Tf 12.776 0 Td [(e + )]TJ/F25 11.9552 Tf 12.777 0 Td [( )]TJ/F25 11.9552 Tf 12.776 0 Td [( c d + c r + h p g ,iii E [ Y ] max f 0, c i + + a = b + )]TJ/F25 11.9552 Tf 12.186 0 Td [( c d + c r = +1 )]TJ/F25 11.9552 Tf 12.186 0 Td [( )]TJ/F25 11.9552 Tf 12.185 0 Td [( c d + c r g ,andiv E [ Y ] max f 0, c i + + a = b + )]TJ/F25 11.9552 Tf 10.093 0 Td [( c d + c r = )]TJ/F39 11.9552 Tf 10.092 0 Td [(d + )]TJ/F25 11.9552 Tf 10.092 0 Td [( )]TJ/F25 11.9552 Tf 10.093 0 Td [( c d + c r g ,respectively.When E [ Y ] satisesi,ii,italsosatisesiiiandiv.Lowerboundson E [ Y ] arenecessaryfor C1,C4,C5,andC6toholdundertheconcavityconditionsgiveninTheorem4.1. Proof. KKTequationtosolveforthetotalallocationquantityinC1canberewritten as E [ Y p )]TJ/F39 11.9552 Tf 12.207 0 Td [(e )]TJ/F39 11.9552 Tf 12.207 0 Td [(v + r )]TJ/F25 11.9552 Tf 12.207 0 Td [( c d + c r F Y R + D + P ]+2 C = E [ Y ] p )]TJ/F39 11.9552 Tf 12.207 0 Td [(e )]TJ/F39 11.9552 Tf 12.207 0 Td [(v + r )]TJ/F22 11.9552 Tf 12.011 0 Td [( c d + c r + h p + c i )]TJ/F39 11.9552 Tf 12.011 0 Td [(v )]TJ/F39 11.9552 Tf 12.012 0 Td [(v h )]TJ/F39 11.9552 Tf 12.011 0 Td [(a = b .Rewriting as E [ Y ]+ )]TJ/F25 11.9552 Tf 12.012 0 Td [( )]TJ/F39 11.9552 Tf 12.011 0 Td [(E [ Y ] and rearranging,therighthandsideisgivenby E [ Y ] p )]TJ/F39 11.9552 Tf 11.596 0 Td [(e )]TJ/F39 11.9552 Tf 11.596 0 Td [(v + r + )]TJ/F25 11.9552 Tf 11.597 0 Td [( )]TJ/F25 11.9552 Tf 11.596 0 Td [( c d + c r + h p )]TJ/F39 11.9552 Tf 11.904 0 Td [(c i + v + v h )]TJ/F39 11.9552 Tf 11.904 0 Td [(a = b )]TJ/F22 11.9552 Tf 11.904 0 Td [( )]TJ/F25 11.9552 Tf 11.904 0 Td [( c d + c r + h p .SincelefthandsideoftheKKTequation isnon-negative,somustbetherighthandside,fromwhichtheprooffollows. AppendixEsummarizestherstorderconditionsandtheKKTcasesundermodels inwhichthesupplyofend-of-lifeproductsdoesnotdependontheacquisitionprice. 4.3.2DeterministicYieldRate Inthissection,weprovidemodelswherethedemanufacturingyieldrateismodeled asadeterministicparameter, y 2 ,1 .Expectedprotfunctionsforthepreliminary testing PTD ,postponedtesting DTD ,andnotesting NTD modelsaregivenin equations4-4,4-5,and4-6,andarestrictlyconcaveintheacquisitionquantitiesas statedinTheorems4.4,4.5,and4.6,respectively: 65

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PTD : PTD R D P = Z yR 0 pxf x dx + Z y R + D yR )]TJ/F39 11.9552 Tf 5.48 -9.683 Td [(p yR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r = y x )]TJ/F25 11.9552 Tf 11.955 0 Td [( yR f x dx + Z y R + D + P y R + D p yR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r = y yD + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F22 11.9552 Tf 11.955 0 Td [( c d + c r = y x )]TJ/F25 11.9552 Tf 11.955 0 Td [( y R + D f x dx + Z 1 y R + D + P y )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(pR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r = y D + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F22 11.9552 Tf 11.955 0 Td [( c d + c r = y P f x dx + Z y R + D + P 0 v )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y R + D + P )]TJ/F39 11.9552 Tf 11.955 0 Td [(x f x dx )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z 1 y R + D + P r )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(x )]TJ/F25 11.9552 Tf 11.955 0 Td [( y R + D + P f x dx )]TJ/F25 11.9552 Tf 11.955 0 Td [( h p P )]TJ/F22 11.9552 Tf 11.956 0 Td [( R + D + P )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b + c i )]TJ/F39 11.9552 Tf 11.955 0 Td [(v )]TJ/F25 11.9552 Tf 11.955 0 Td [( y )]TJ/F39 11.9552 Tf 11.955 0 Td [(v h R + D + P )]TJ/F25 11.9552 Tf 11.956 0 Td [( c d R + D + c r R -4 DTD : DTD R D P = Z yR 0 pxf x dx + Z y R + D yR )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(p yR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r = y x )]TJ/F25 11.9552 Tf 11.956 0 Td [( yR f x dx + Z y R + D + P y R + D p yR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r = y yD + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F22 11.9552 Tf 11.955 0 Td [( c d + c i = y )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r = y x )]TJ/F25 11.9552 Tf 11.955 0 Td [( y R + D f x dx + Z 1 y R + D + P y )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(pR + p )]TJ/F39 11.9552 Tf 11.956 0 Td [(d )]TJ/F39 11.9552 Tf 11.956 0 Td [(c r = y D + p )]TJ/F39 11.9552 Tf 11.956 0 Td [(e )]TJ/F22 11.9552 Tf 11.955 0 Td [( c d + c i = y )]TJ/F25 11.9552 Tf 11.955 0 Td [( c r = y P f x dx + Z y R + D + P 0 v )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y R + D + P )]TJ/F39 11.9552 Tf 11.955 0 Td [(x f x dx )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z 1 y R + D + P r )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(x )]TJ/F25 11.9552 Tf 11.955 0 Td [( y R + D + P f x dx )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p P )]TJ/F22 11.9552 Tf 11.956 0 Td [( R + D + P )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b )]TJ/F39 11.9552 Tf 11.955 0 Td [(v )]TJ/F25 11.9552 Tf 11.956 0 Td [( y )]TJ/F39 11.9552 Tf 11.956 0 Td [(v h R + D + P )]TJ/F22 11.9552 Tf 11.955 0 Td [( c i + c d R + D )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r R -5 NTD : NTD R D P = Z yR 0 pxf x dx + Z y R + D yR )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(pyR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r = y x )]TJ/F39 11.9552 Tf 11.955 0 Td [(yR f x dx + Z y R + D + P y R + D )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(pyR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r = y yD + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F22 11.9552 Tf 11.955 0 Td [( c d + c r = y x )]TJ/F39 11.9552 Tf 11.955 0 Td [(y R + D f x dx + Z 1 y R + D + P y )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(pR + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r = y D + p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F22 11.9552 Tf 11.956 0 Td [( c d + c r = y P f x dx + Z y R + D + P 0 v )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(y R + D + P )]TJ/F39 11.9552 Tf 11.955 0 Td [(x f x dx )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z 1 y R + D + P r )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(x )]TJ/F39 11.9552 Tf 11.955 0 Td [(y R + D + P f x dx )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p P )]TJ/F22 11.9552 Tf 11.955 0 Td [( R + D + P )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b )]TJ/F39 11.9552 Tf 11.956 0 Td [(v )]TJ/F25 11.9552 Tf 11.955 0 Td [( y )]TJ/F39 11.9552 Tf 11.955 0 Td [(v h R + D + P )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d R + D )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r R -6 66

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Theorem4.4. PTD R D P isstrictlyconcavein R D and P ifandonlyifthe followingconditionsholdforagiven y 2 ,1 :i y p )]TJ/F39 11.9552 Tf 11.666 0 Td [(e )]TJ/F30 11.9552 Tf 11.666 9.684 Td [()]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( y + )]TJ/F25 11.9552 Tf 11.667 0 Td [( )]TJ/F39 11.9552 Tf 11.666 0 Td [(y c d + c r > y v )]TJ/F39 11.9552 Tf 12.431 0 Td [(r ,i.e.,satisfyingthereusablepartdemandfromthestockofend-of-life productsismoreprotablethansalvagingthepart,earningthesalvagerevenueand incurringthepenaltyforthelostsale,whichleadsto y > y LB PTD = )]TJ/F25 11.9552 Tf 11.991 0 Td [( c d + c r = )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [( p )]TJ/F39 11.9552 Tf -457.446 -23.908 Td [(e )]TJ/F39 11.9552 Tf 11.1 0 Td [(c d )]TJ/F39 11.9552 Tf 11.1 0 Td [(c r )]TJ/F39 11.9552 Tf 11.1 0 Td [(v + r + )]TJ/F25 11.9552 Tf 11.101 0 Td [( c d + c r ,requiring p )]TJ/F39 11.9552 Tf 11.1 0 Td [(e )]TJ/F39 11.9552 Tf 11.1 0 Td [(c d )]TJ/F39 11.9552 Tf 11.101 0 Td [(c r > v )]TJ/F39 11.9552 Tf 11.1 0 Td [(r ,and y > y LB PTD ,i.e., thedemanufacturingyieldratetoexceedacriticalpercentagegivenbytheproportion oftheoperationalcostsincurredforafalsepositiveduetoincorrectlyidentifyingitas recoverabletothetotalcostincurredforacorrectlyidentiedtruepositiveinadditionto theformercost,ii e + c d + c r = y > d + c r = y ,i.e.,themarginalcostofpostponing thedisassemblyoperationisgreaterthanthemarginalcostofpostponingtherecovery operation,andiii d + c r = y > 0 ,i.e.,themarginalcostofpostponingtherecovery operationispositive. Proof. Firstorderoptimalityconditionsaregivenbelow: PTD 0 R = )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( y p )]TJ/F39 11.9552 Tf 11.956 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( c d + c r F y R + D + P )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( y e )]TJ/F39 11.9552 Tf 11.955 0 Td [(d + c d F y R + D )]TJ/F22 11.9552 Tf 11.955 0 Td [( yd + c r F yR )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = b + y p )]TJ/F39 11.9552 Tf 11.956 0 Td [(v + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( c d + c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h + a = b PTD 0 D = )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( c d + c r F y R + D + P )]TJ/F30 11.9552 Tf 11.956 9.684 Td [()]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y e )]TJ/F39 11.9552 Tf 11.955 0 Td [(d + c d F y R + D )]TJ/F22 11.9552 Tf 11.956 0 Td [(2 R + D + P = b + y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( c d + c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h + a = b PTD 0 P = )]TJ/F30 11.9552 Tf 11.956 9.684 Td [()]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( c d + c r F y R + D + P )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = b + y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( c d + c r + h p )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h + a = b TheHessianmatrixhasthesameformgivenintheproofofTheorem3.1with A = y )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y p )]TJ/F39 11.9552 Tf 12.158 0 Td [(e )]TJ/F39 11.9552 Tf 12.158 0 Td [(v + r )]TJ/F25 11.9552 Tf 12.158 0 Td [( c d + c r f y R + D + P +2 = b B = y )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( y e )]TJ/F39 11.9552 Tf 12.158 0 Td [(d + c d f y R + D ,and C = y )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( y + c r f yR ,fromwhichtheprooffollows. 67

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Theorem4.5. DTD R D P isstrictlyconcavein R D and P ifandonlyifthe followingconditionsholdforagiven y 2 ,1 :i y p )]TJ/F39 11.9552 Tf 11.613 0 Td [(e )]TJ/F39 11.9552 Tf 11.613 0 Td [(c d )]TJ/F39 11.9552 Tf 11.613 0 Td [(c i )]TJ/F30 11.9552 Tf 11.613 9.684 Td [()]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y + )]TJ/F25 11.9552 Tf 11.613 0 Td [( )]TJ/F39 11.9552 Tf -458.701 -23.908 Td [(y c r > y v )]TJ/F39 11.9552 Tf 12.139 0 Td [(r ,i.e.,satisfyingthereusablepartdemandfromthestockofend-of-life productsismoreprotablethansalvagingthepart,earningthesalvagerevenueand incurringthepenaltyforthelostsale,whichleadsto y > y LB DTD = )]TJ/F39 11.9552 Tf 5.479 -9.683 Td [(c d + c i + )]TJ/F25 11.9552 Tf -424.356 -23.908 Td [( c r = )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [( p )]TJ/F39 11.9552 Tf 11.429 0 Td [(e )]TJ/F39 11.9552 Tf 11.43 0 Td [(c r )]TJ/F39 11.9552 Tf 11.43 0 Td [(v + r + )]TJ/F25 11.9552 Tf 11.429 0 Td [( c r ,requiring p )]TJ/F39 11.9552 Tf 11.43 0 Td [(e )]TJ/F39 11.9552 Tf 11.43 0 Td [(c r )]TJ/F39 11.9552 Tf 11.43 0 Td [(c d )]TJ/F39 11.9552 Tf 11.43 0 Td [(c i > v )]TJ/F39 11.9552 Tf 11.43 0 Td [(r ,and y > y LB DTD ,i.e.,thedemanufacturingyieldratetoexceedacriticalpercentagegivenby theproportionoftheinspectionandoperationalcostsincurredforafalsepositivedueto incorrectlyidentifyingitasrecoverabletothetotalcostincurredforacorrectlyidentied truepositiveinadditiontotheformercost,ii e + c d + c i = y + c r = y > d + c r = y ,i.e., themarginalcostofpostponingthedisassemblyoperationisgreaterthanthemarginal costofpostponingtherecoveryoperation,andiii d + c r = y > 0 ,i.e.,themarginalcost ofpostponingtherecoveryoperationispositive. Proof. Firstorderoptimalityconditionsaregivenbelow: DTD 0 R = )]TJ/F30 11.9552 Tf 11.956 9.684 Td [()]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i )]TJ/F25 11.9552 Tf 11.956 0 Td [( c r F y R + D + P )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y e )]TJ/F39 11.9552 Tf 11.956 0 Td [(d + c d + c i F y R + D )]TJ/F22 11.9552 Tf 11.955 0 Td [( yd + c r F yR )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = b + y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h + a = b DTD 0 D = )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i )]TJ/F25 11.9552 Tf 11.955 0 Td [( c r F y R + D + P )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y e )]TJ/F39 11.9552 Tf 11.955 0 Td [(d + c d + c i F y R + D )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = b + y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F25 11.9552 Tf 11.956 0 Td [( c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h + a = b DTD 0 P = )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i )]TJ/F25 11.9552 Tf 11.955 0 Td [( c r F y R + D + P )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = b + y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p )]TJ/F25 11.9552 Tf 11.955 0 Td [( c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h + a = b TheHessianmatrixhasthesameformgivenintheproofofTheorem3.1with A = y )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [( y p )]TJ/F39 11.9552 Tf 11.928 0 Td [(e )]TJ/F39 11.9552 Tf 11.928 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.928 0 Td [(c d )]TJ/F39 11.9552 Tf 11.928 0 Td [(c i )]TJ/F25 11.9552 Tf 11.928 0 Td [( c r f y R + D + P +2 = b B = y )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [( y e )]TJ/F39 11.9552 Tf 11.928 0 Td [(d + c d + c i f y R + D ,and C = y )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [( y + c r f yR ,fromwhichtheprooffollows. 68

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Theorem4.6. NTD R D P isstrictlyconcavein R D and P ifandonlyifthe followingconditionsholdforagiven y 2 ,1 :i y p )]TJ/F39 11.9552 Tf 12.882 0 Td [(e )]TJ/F39 11.9552 Tf 12.882 0 Td [(c d )]TJ/F39 11.9552 Tf 12.882 0 Td [(c r > y v )]TJ/F39 11.9552 Tf 12.882 0 Td [(r i.e.,satisfyingthereusablepartdemandfromthestockofend-of-lifeproductsismore protablethansalvagingthepart,earningthesalvagerevenueandincurringthepenalty forthelostsale,whichleadsto y > y LB NTD = c d + c r = p )]TJ/F39 11.9552 Tf 12.981 0 Td [(e )]TJ/F39 11.9552 Tf 12.981 0 Td [(v + r ,requiring p )]TJ/F39 11.9552 Tf 12.436 0 Td [(e )]TJ/F39 11.9552 Tf 12.436 0 Td [(c d )]TJ/F39 11.9552 Tf 12.437 0 Td [(c r > v )]TJ/F39 11.9552 Tf 12.436 0 Td [(r ,and y > y LB NTD ,i.e.,thedemanufacturingyieldratetoexceed acriticalpercentagegivenbytheproportionoftheoperationalcostsincurredforan unrecoverableparttothetotalcostincurredforarecoverablepartinadditiontothe formercost,ii e + c d + c r = y > d + c r = y ,i.e.,themarginalcostofpostponingthe disassemblyoperationisgreaterthanthemarginalcostofpostponingtherecovery operation,andiii d + c r = y > 0 ,i.e.,themarginalcostofpostponingtherecovery operationispositive. Proof. Firstorderoptimalityconditionsaregivenbelow: NTD 0 R = )]TJ/F22 11.9552 Tf 11.956 0 Td [(2 R + D + P = b )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(y p )]TJ/F39 11.9552 Tf 11.956 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.956 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r F y R + D + P )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(y e )]TJ/F39 11.9552 Tf 11.955 0 Td [(d + c d F y R + D )]TJ/F22 11.9552 Tf 11.955 0 Td [( yd + c r F yR + y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.956 0 Td [(c d )]TJ/F39 11.9552 Tf 11.956 0 Td [(c r + v + v h + a = b NTD 0 D = )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = b )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r F y R + D + P )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(y e )]TJ/F39 11.9552 Tf 11.955 0 Td [(d + c d F y R + D + y p )]TJ/F39 11.9552 Tf 11.956 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r + v + v h + a = b NTS 0 P = )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 R + D + P = b )]TJ/F30 11.9552 Tf 11.955 9.684 Td [()]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r F y R + D + P + y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r + v + v h + a = b TheHessianmatrixhasthesameformgivenintheproofofTheorem3.1with A = y )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(y p )]TJ/F39 11.9552 Tf 10.921 0 Td [(e )]TJ/F39 11.9552 Tf 10.921 0 Td [(v + r )]TJ/F39 11.9552 Tf 10.922 0 Td [(c d )]TJ/F39 11.9552 Tf 10.921 0 Td [(c r f y R + D + P +2 = b B = y )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(y e )]TJ/F39 11.9552 Tf 10.921 0 Td [(d + c d f y R + D and C = y )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(y + c r f yR ,fromwhichtheprooffollows. ThesolutionapproachusingKKTconditionsisstillapplicable.Tables4-8,4-9, and4-10listtheapplicableKKTcasesunderthegeneraldemanddistribution.Similarly, 69

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Tables4-11,4-12,and4-13listthesolutionsfortheKKTcasesunderuniformly distributeddemand,forwhichexplicitsolutionsareavailable. Table4-8.PTD:KKTcasesunderdeterministicdemanufacturingyieldrateandgeneral demanddistribution. CaseEquationstosolve C1 F yR = yd = yd + c r F y R + D = y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p = y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.962 0 Td [( c d + c r F y R + D + P +2 R + D + P = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.963 0 Td [( c d + c r + h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h + a = b C4 F yR = ye + h p = ye + c d + c r y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.962 0 Td [( c d + c r F y R + P +2 R + P = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.963 0 Td [( c d + c r + h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h + a = b C5 y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r F yR +2 R = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.962 0 Td [( c d + c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b C6 F yR = yd = yd + c r y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.963 0 Td [( c r F y R + D +2 R + D = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.963 0 Td [( c d + c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b Table4-9.DTD:KKTcasesunderdeterministicdemanufacturingyieldrateandgeneral demanddistribution. CaseEquationstosolve C1 F yR = yd = yd + c r F y R + D = y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p = y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d + c i y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i )]TJ/F11 9.9626 Tf 9.962 0 Td [( c r F y R + D + P +2 R + D + P = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F11 9.9626 Tf 9.963 0 Td [( c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b C4 F yR = ye + h p = ye + c d + c i + c r y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i )]TJ/F11 9.9626 Tf 9.962 0 Td [( c r F y R + P +2 R + P = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F11 9.9626 Tf 9.963 0 Td [( c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b C5 y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r F yR +2 R = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F11 9.9626 Tf 9.963 0 Td [( c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b C6 F yR = yd = yd + c r y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.963 0 Td [( c r F y R + D +2 R + D = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F11 9.9626 Tf 9.963 0 Td [( c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b Table4-10.NTD:KKTcasesunderdeterministicdemanufacturingyieldrateandgeneral demanddistribution. CaseEquationstosolve C1 F yR = yd = yd + c r F y R + D = y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p = y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r F y R + D + P +2 R + D + P = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r + v + v h + a = b C4 F yR = ye + h p = ye + c d + c r y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r F y R + P +2 R + P = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r + v + v h + a = b C5 y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r F yR +2 R = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r + v + v h + a = b C6 F yR = yd = yd + c r y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r F y R + D +2 R + D = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r + v + v h + a = b 70

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Table4-11.PTD:KKTcasesunderdeterministicdemanufacturingyieldrateanduniform U demand. CaseEquationstosolve C1 F yR = yd = yd + c r F y R + D = y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p = y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d F y R + D + P = y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.962 0 Td [( c d + c r + h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b = y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.963 0 Td [( c d + c r +2 U = b y C4 F yR = ye + h p = ye + c d + c r F y R + P = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.963 0 Td [( c d + c r + h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.963 0 Td [( c d + c r +2 U = b y C5 F yR = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.963 0 Td [( c d + c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b = y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r +2 U = b y C6 F yR = yd = yd + c r F y R + D = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.962 0 Td [( c d + c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h + a = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.962 0 Td [( c r +2 U = b y Table4-12.DTD:KKTcasesunderdeterministicdemanufacturingyieldrateanduniform U demand. CaseEquationstosolve C1 F yR = yd = yd + c r F y R + D = y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p = y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d + c i F y R + D + P = y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F11 9.9626 Tf 9.962 0 Td [( c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h + a = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i )]TJ/F11 9.9626 Tf 9.963 0 Td [( c r +2 U = b y C4 F yR = ye + h p = ye + c d + c i + c r F y R + P = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F11 9.9626 Tf 9.963 0 Td [( c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i )]TJ/F11 9.9626 Tf 9.963 0 Td [( c r +2 U = b y C5 F yR = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F11 9.9626 Tf 9.962 0 Td [( c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h + a = b = y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r +2 U = b y C6 F yR = yd = yd + c r F y R + D = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F11 9.9626 Tf 9.963 0 Td [( c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h + a = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.962 0 Td [( c r +2 U = b y Table4-13.NTD:KKTcasesunderdeterministicdemanufacturingyieldrateanduniform U demand. CaseEquationstosolve C1 F yR = yd = yd + c r F y R + D = y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p = y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d F y R + D + P = y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c r + v + v h + a = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r +2 U = by C4 F yR = ye + h p = ye + c d + c r F y R + P = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r + v + v h + a = b = y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r +2 U = by C5 F yR = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r + v + v h + a = b = y p )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r +2 U = by C6 F yR = yd = yd + c r F y R + D = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r + v + v h + a = b = y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r +2 U = by Alsosimilartothemodelsunderstochasticyield,KKTequationscanbeused toderivelowerboundsonthedemanufacturingyieldratethatarenecessaryforC1, 71

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C4,C5,andC6toholdseeAppendixD.Noticethatthelowerboundsobtainedfrom theconcavityconditionarestrongerthanthoseobtainedfromtheKKTequations if a = b + c i )]TJ/F39 11.9552 Tf 13.034 0 Td [(v )]TJ/F39 11.9552 Tf 13.035 0 Td [(v h > 0 intheacquisition-price-dependentsupplymodels,andif c + c i )]TJ/F39 11.9552 Tf 10.923 0 Td [(v )]TJ/F39 11.9552 Tf 10.923 0 Td [(v h > 0 intheacquisition-price-independentsupplymodels.Theseinequalities aremeaningfulsinceotherwisesalvagerevenueswillexceedthecostofacquisition. 4.4ComputationalExperiments Inournumericalstudy,wecomparethepreliminaryandthepostponedtesting scenarioswitheachother,andalsowithabenchmarkscenarioinwhichnodiagnostic testingisconducted,andinvestigatethevalueofthediagnostictesting.Furthermore, wereportthesensitivityoftheprotsobtainedundereachscenariotoproblem parameters.Ourparametersetissimilartotheoneweusedinourrstresearch withsomemodications.Wesetawiderrangeforthelowandhighlevelsofthetesting costsinordertobeabletoinvestigateitsimpact.Duetoadditionalparameters wereducethenumberofparametersbyxingsomeofthevaluesofparameters. Furthermore,weonlyreporttheresultsoftheexperimentsunderuniformlydistributed demand,andstochasticdemanufacturingyieldrate.Inournumericalstudy,wehave 2,880probleminstances.Table3-6listsourparameterset: Table4-14.Parametersetforthetestingscenariosinone-periodnewsvendormodels. pabcc i Y CV Y h p Intensity c d + c r edr v v h 501030300.10.750.850.850.00611.5 d 203052520 ,10 10.800.900.903 b 30 0.850.950.956 r 0.9011 0.95 %of:app ---pc d + c r p pepp Inournumericalexperiments,weobservedthattheprotabilityofadiagnostictest depends,rst,onitsaccuracylevel,andsecond,onthestageofthediagnostictestifit isgoingtobeconducted,coupledwiththesupplytypeoftheend-of-lifeproducts,andof alesserdegreeonthedemanufacturingyieldrate,testingcost,andotherparameters. 72

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Modelswithimperfectdiagnostictestsyieldlowerprotscomparedtothenotesting scenario,withpostponedtestingperformingworsethanthepreliminarytestingscenario, atallexpecteddemanufacturingyieldrates.Astheexpecteddemanufacturingyield rateincreasesalongwiththeaccuracyofthetest,eachmodelyieldssimilarprotsas expected.Whenthediagnostictestaccuracyis100%,preliminarytestingscenarioisthe bestintermsoftheexpectedtotalprots,followedbythepostponedtesting,regardless ofthetestingcosts,environmentalfees,andtheoperationalcostbalanceweconsidered inournumericalstudy. However,whenthesupplyquantityoftheend-of-lifeproductsdoesnotdependon theacquisitionprice,thentheno-testingscenariosuffersunderverylowlevelsofthe expecteddemanufacturingyieldratewhentheaccuracyofthediagnostictestingis sufcientlyhighforthetestingscenariostolterouttheunrecoverableparts.FiguresF-1 andF-2inAppendixFshowthetotalacquisitionquantitiesunderbothtypesofsupply, whereunderacquisition-price-independentsupply,itcanbeseenthattheno-testing scenariocannotacquireasmanyend-of-lifeproductsasitcouldunderthecasewhenit hasthefreedomtochoosetheacquisitionpriceseeFigureF-3inAppendixF. Inthefollowingthreecases,wedemonstrateourothermajorobservations,and indicatepossibleapplicationareasfromreallifebyconsideringrelativevaluesofthe diagnostictestingaccuracyandcosts. Case1:Equaltestingaccuraciesandcosts. Thiscaseemphasizestheimportanceofstagingthediagnostictestonthe protabilityofthedemanufacturingbusiness,i.e.,whentoperformthetestifitis goingtobeconducted:immediatelyafteracquisitionandbeforeallocationorafter thedisassemblyoperationiscompleted.Whentestingaccuraciesandcostsareequal, postponingthediagnostictestingisinferiortoprioritizingitintermsofexpectedprots. Whereaspreliminarytestingeliminatessomeoftheend-of-lifeproductscontaining unrecoverableparts,postponedtestingdelaystheinspectionuntilafterthedisassembly 73

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operationiscompleted.Consequently,protsarereducedduetounnecessarily incurringtheenvironmentalfeesforend-of-lifeproductskeptinstockand/orthecostof thedisassemblyoperationforitemswithuncertainconditionseeFigures4-3and4-4. Therefore,ifadiagnostictestcanbeconductedatanydemanufacturingstageatthe samecost,itismoreprotabletodoituponreceipt,andbeforeincurringanyadditional costsofdemanufacturingandenvironmentalfees. A B C D E F G H I Figure4-3.Comparisonofexpectedtotalprotsunderacquisition-price-dependent supply,stochasticyieldmodels, a b =%,30% c i = L c d + c r = L A,D,G = =0.85 .B,E,H = =0.90 .C,F,I = =1 74

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A B C D E F G H I Figure4-4.Comparisonofexpectedtotalprotsunderacquisition-price-independent supply,stochasticyieldmodels, c i = L c d + c r = L A,D,G = =0.85 .B,E,H = =0.90 .C,F,I = =1 Case2:Equaltestingcostsbutperfectaccuracyinpostponedtesting. Thiscaseemphasizestheimportanceofaccessibilitytothereusablepart,bywhich aperfectdiagnosismaybepossible,andreferstoreallifecaseswhereadiagnostic testingcanbeperformedeitherbeforeorafterthedisassemblyoperation,buthas perfectaccuracyifitcanbeconducteddirectlyontheextractedpart.Forinstance, inspectionwithadiagnostictestingtoolstrengthenedwithavisualinspectionthatmay 75

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notbepossiblebeforeextractionofthepartfromtheend-of-lifeproductisanexampleto thiscase.Inthiscaseitismoreeconomicaltoperformthedisassembly,andpostpone thediagnostictestuntilaftercompletionofthedisassemblyoperationseeFigures4-5 and4-6.However,weremarkthatforthissituationtooccur,thediagnostictestingon theextractedpartshouldhaveperfectaccuracy,i.e.,theclassicationerrorsofboth typesshouldbezero. A B C D E F G H I Figure4-5.Comparisonofexpectedtotalprotsunderacquisition-price-dependent supply,stochasticyieldmodels, a b =%,30% c i = L c d + c r = L DT = DT =1 .A,D,G PT = PT =0.85 .B,E,H PT = PT =0.90 C,F,I PT = PT =0.95 76

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A B C D E F G H I Figure4-6.Comparisonofexpectedtotalprotsunderacquisition-price-independent supply,stochasticyieldmodels, c i = L c d + c r = L DT = DT =1 A,D,G PT = PT =0.85 .B,E,H PT = PT =0.90 C,F,I PT = PT =0.95 ItcanbeseeninFigure4-5thatwhenthesupplyquantityoftheend-of-lifeproducts dependsontheacquisitionprice,thediscrepancybetweentheexpectedprotsgets largerasthepreliminarytestingscenario'saccuracydecreases.Inthiscase,preliminary testingyieldstheworstprotsamongallscenarios,whereaspostponedtestingperforms betterthanthescenariowithnotesting.Astheaccuracyofthepreliminarytesting 77

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increases,theprotdiscrepancystartstodiminish,wherepostponedtestingstill performsthebest.ItcanbeseeninFigure4-6thatwhentheacquisitionpricedoesnot affectthesupplyquantityoftheend-of-lifeproducts,similarobservationshold,where allmodelsperformsslightlybetterthantheydoundertheacquisition-price-dependent supply. Therefore,ifitcoststhesametoconductthediagnostictestingatanystage,but withperfectaccuracyontheextractedpart,thenitisbettertopostponethetestinguntil afterthedisassemblyoperationiscompleted. Case3:Highertestingcostswithperfectaccuracyinpostponedtesting. Thiscaseemphasizestheimportanceofthestagingandthetypeofthediagnostic testing,wherepreliminarytestingcanbeconsideredasapre-sortingprocedureto classifytheend-of-lifeproductssuchasvisualexamination,andpostponedtestingcan beconsideredasamoresophisticatedformalproceduresuchastestingusingspecial diagnostictool. Whentheperfectpostponedtestingcostishighcomparedtotheassociatedcost ofasignicantlyimperfectpreliminarytesting,theattractivenessoftheformerisstill signicantseeFigures4-7A,4-7D,4-7G,4-8A,4-8D,and4-8G.However,theprot discrepancybetweenthepostponedtestingandno-testingscenariosdiminishesdueto thehighercostoftestingontheextractedpart,i.e.,thepostponedtesting. Astheaccuracyofthepreliminarytestingincreases,theperformancesofall scenariosgetclosertoeachotherintermsoftheexpectedprots.Undersufciently highlevelsofpreliminarytestingaccuracy,postponedtesting'shighercostcompared tothelowercostofthepreliminarytesting,inadditiontothedisassemblycost,results inlowerexpectedtotalprotsunderpostponedtestingascomparedtothoseunder preliminarytestingsee,forinstance,Figures4-7C,4-7F,4-7I,4-8C,4-8F,and4-8I. Therefore,whenthecostofaperfectpostponedtestishigherthanthecostofan imperfectpreliminarytest,itismoreprotabletopostponethediagnostictestifthe 78

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preliminarytestingaccuracyisverylow,andifthedemanufacturerdoesnothavethe exibilitytosetanacquisitionpriceontheend-of-lifeproducts,i.e.,iftheacquisition priceisnotadecisionvariableseeFigure4-7.Ontheotherhand,ifthepreliminary testingissignicantlyaccurate,andifthesupplyquantityofend-of-lifeproductsis independentoftheacquisitionprice,itismoreprotabletoprioritizethediagnostictest seeFigure4-8. A B C D E F G H I Figure4-7.Comparisonofexpectedtotalprotsunderacquisition-price-dependent supply,stochasticyieldmodels, a b =%,30% c i PT c i DT = L H c d + c r = L DT = DT =1 .A,D,G PT = PT =0.85 B,E,H PT = PT =0.90 .C,F,I PT = PT =0.95 79

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A B C D E F G H I Figure4-8.Comparisonofexpectedtotalprotsunderacquisition-price-independent supply,stochasticyieldmodels, a b =%,30% c i PT c i DT = L H c d + c r = L DT = DT =1 .A,D,G PT = PT =0.85 B,E,H PT = PT =0.90 .C,F,I PT = PT =0.95 80

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CHAPTER5 MYOPICTWO-PERIODNEWSVENDORPOLICIESFORDEMANUFACTURING UNDERLIFECYCLECONSIDERATIONOFREUSABLEPARTS 5.1BackgroundandMotivation Marketlifecycleofusedandrecoveredpartsonsomecomplexend-of-lifeproducts maybelimitedduetochangesinthedesignoftheproductsandpossibleupdates tothedesignofthepartthatrenderdemanufacturingofsuchpartsforsaledifcult. Furthermore,productandpartdesignchangesareusuallyunknownuntiltheyare announcedpubliclybythemanufacturerduetomarketcompetition.Forinstance,Apple Inc.,acorporationthatmanufacturesandsellsconsumerelectronicsamongother productssuchassoftware,announcessuchdesignchangesorupdatestopublicatthe WorldWideDevelopersConference,untilwhichdetailspertainingtothenewgeneration ofaproductand/orthereusablepartunderconsiderationonthenewproductremains unknown.Therefore,ademanufacturerwhoacquiresaparticularend-of-lifeproductto demanufactureandsellaparticularreusablepartonitduringapredeterminedsales periodmayhavetoconsiderhowmuchtoacquireandstockforthecurrentperiodas wellaswhattodowithhisstockofolderversionofaproductinthefutureperiodafter aproductreceivesadesignchange.Thisisbecauseoftheuncertaintyassociated withthepartonthenewproduct,whichmayormaynotreceiveadesignchangeas well,andtheextentofthedesignchangethatmaymaketheoldpartcostlyordifcult todemanufacture.Thedecisionsofhowmuchtoacquireandstock,consideringthe currentandthefutureperiods,maybequitecomplicatedundertheaforementioned uncertaintyassociatedwiththenewgenerationofthepartontheend-of-lifeproduct, coupledwiththedemanduncertaintyinbothperiods. Insomecases,theproductmaybecompletelyrevisedinthenextperiodwith adifferentpartthatmakestheoldversionofthepartobsolete.Insomecases, however,thepartontheend-of-lifeproductmaybeunchangedorreceiveminor designchanges,makingitpossibleforthedemanufacturertocontinuehisoperations, 81

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andmakeuseofhisexistinginventoryofitemsfromtheoldergeneration.When thereisapartcommonalitybetweentheoldandnewgenerationsofanend-of-life productwithpossibleminormodicationstotheoldgenerationofthereusablepart, thedemanufacturermayhavetoincurthecostofmoreexpensivedemanufacturing operationsduetorequiredchangesinthedemanufacturingprocesses.Evenifthepart underconsiderationisunchangedfromoneperiodtothenext,end-of-lifeproductsthat areacquiredlaterintheirlife-cyclemayhavemoretearandwearduetopassageof timeorusage,increasingthedemanufacturingcostsoftheitemsacquiredinthelater periods.Therefore,thedemanufacturermayhavetoconsiderthemorecostlyoptionof demanufacturinginthesecondperiodascomparedtousingtheexistingstockofitems andincurringtheassociatedinventoryholdingcosts. Existenceoftheenvironmentalfeesmayalsodiscouragethedemanufacturerfrom carryingtheremainingstockofend-of-lifeproductsintothenextperiodbecauseofthe recurringenvironmentalfeesinthenextperiod.Consequently,thedemanufacturer maychoosetoeitherprocesstheexistingend-of-lifeproductsincurringtheassociated demanufacturingcosts,andcarryonlytherecoveredpartsintothenextperiod,incurring theinventoryholdingcostsortoselltheleft-overend-of-lifeproductstoamaterial recyclerfortheirmaterialcontent,earningthesalvagerevenues. Duetodiscusseduncertaintiesinthetwoperiods,itisanalyticallycomplicatedto solvethetwo-periodproblem,andmaynotbemeaningfulfromapracticalpointofview forsomeproductssuchasconsumerproductsincludingelectronicsthathaveshortlife cyclesandaresubjecttofrequentdesignchanges.Instead,thedemanufacturermay considerthecurrentperiodasastand-aloneone,inwhichhedecidesonthequantity ofend-of-lifeproductstoacquire,andthequantitiesofend-of-lifeproductstoallocate amongthestockpointsuponperformingthenecessarydemanufacturingoperations tomaximizehisprotsintherstperiod,assumingnoinventorycarry-overtothenext period.Inthesecondperiod,hedecidesonwhethertocontinuedemanufacturingthe 82

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partbasedonthecostparameters,andwhetherthereispartcommonalitybetween theoldandthenewgenerationsoftheproduct.Ifthepartonthenewgenerationofthe productisthesameastheoneontheoldgenerationorcanbedemanufacturedwith minorchangestothedemanufacturingprocesssothatprotmarginscanbegenerated, thenthedemanufacturermaydecideonthequantitiesofacquisition,andtheallocation oftheacquireditemsconsideringtheremainingstockfromtherstperiod. Inthispartofthedissertation,weinvestigatethreepracticallyimplementable myopicpoliciestomaximizethedemanufacturer'sexpectedprotsoveratwo-period horizon,inwhich,therstperiodisconsideredasastand-aloneperiodinanewsvendor typeofsetting.Intherstpolicy,weassumethatthesecondperiodisalsostand-alone periodsimilartotherstperiod,andthedemanufacturerdoesnotcarrytheexisting stockofitemsintothesecondperiod.Inthesecondpolicy,thedemanufacturer optimizeshisexpectedprotsintherstperiodassumingnoinventorycarry-over intothesecondperiod,butinrealityhecarriestheleft-overinventoryofrecoveredparts intothesecondperiod,salvaginganyexistingend-of-lifeproductsfromthesecond period.Thethirdpolicyissimilartothesecondpolicywhere,thistime,theend-of-life productsremaininginstockattheendoftherstperiodaredemanufactured,andare carried-overintothenextperiodalongwiththealreadyrecoveredparts. 5.2ProblemDenition Theproblemsettingweconsiderinthispartofthedissertationisquitesimilartothe onesweconsiderinthepreviouschapterswithsomechangesasdescribedbelow.We considertwoperiods,labeledby t =1,2 .Allcostanddemandparametersareindexed bytheperiodtheyareapplicablein.Similartoourpreviousresearchproblemswhere thesupplyquantityofend-of-lifeproductsisindependentoftheacquisitionprice,the demanufactureracquiresaparticularend-of-lifeproductfromtheend-usermarketata pre-determinedacquisitionprice, c t ,withasinglereusablepartonit.Weassumethat thereisnoinitialinventoryofitemsatthebeginningoftherstperiod. 83

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Afteracquisition,thedemanufacturerconductsadiagnostictestonallacquired end-of-lifeproductsataunitcostof c i t ,andbeforestockingthem.End-of-lifeproducts containingunrecoverablepartscanbesoldtoamaterialrecyclerearningrevenuesfrom boththepart, v t ,andthehulk, v h t pereachitem.Inordertosimplifytheanalysis,we assumethatthediagnostictestyieldstrueconclusions,i.e.,classicationerrordoes notexist.Furthermore,weassumethatthedemanufacturingyieldrate, y t 2 ,1 ,is deterministic.Thestochasticdemanufacturingyieldrate,despitebeingmorerealistic, waspreviouslyobservedtoaffectonlythesystemprots,resultinginhigheracquisition andallocationquantities,butleavingthemodelbehaviorssimilartothoseunderthe deterministicdemanufacturingyieldrateinourrstresearch.Therefore,wepreferto simplifytheanalysisusingadeterministicdemanufacturingyieldrate. Differentthanourpreviousresearchproblemsettings,thedemanufacturermay allocatetheacquiredend-of-lifeproductsamongtwostockpointsfor:end-of-life productsandrecoveredpartsuponperformingthenecessarydemanufacturing operationstesting,disassembly,andrecovery.Inotherwords,wecombinethe demanufacturingoperationsofdisassemblyandrecovery,andkeepnostocksof disassembledbutnotrecoveredparts,whichsimpliestheanalyticalprocessofderiving quantitiesofinterest.Acquisitionquantitiesofend-of-lifeproductstoallocateamongthe stockpointsforend-of-lifeproductsisdenotedby P t ,andrecoveredpartsisdenotedby R t .Initialinventoryofrecoveredpartsis I R t )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 ,where I R ,0 =0 seeFigure5-1. Figure5-1.Flowofacquiredquantityofend-of-lifeproductsinthesystem. 84

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Aunitenvironmentalfeeof h p t isincurredforeachend-of-lifeproductthatis keptinstock.Thereusableparthastobeextractedfromtheend-of-lifeproductbya disassemblyoperation,andreconditionedbyarecoveryoperationatatotalunitcost, c r t ,afterwhichitisstoredinthestockpointfortherecoveredparts,andissuitablefor saleintheused-partsmarketataunitpriceof p t .Thehulkremainingaftertheextraction ofthereusablepartcanbesoldtothematerialrecyclerataunitrevenueof v h t Thedemandfortherecoveredpartsinaperiod, X t ,israndomwithaknown probabilitydensityfunction, f t x t .Demandsinthetwoperiodsareassumedtobe independent,butnotnecessarilyidentical.Similartoourpreviousstudies,thedemand isrstsatisedfromtherecoveredpartsstockifthereareavailableparts,andfromthe end-of-life-productsstockpointincasequantityofrecoveredpartsisinsufcient.We assumethattheunsatisedportionofthedemandislost,andthedemanufacturerincurs aloss-of-goodwillcost, r t ,foreachunmetdemand. Thedecisionsthattakeplaceattheendoftherst,andatthebeginningofthe secondperiodsdependonthemyopicpolicy.Intherstpolicy,whichwecallas standaloneperiods ,thedemanufacturerdiscardsanyremaininginventoryofend-of-life productsandrecoveredpartsbysellingthemtothematerialrecyclerfortheirmaterial value,earningthesalvagerevenuesfromthehulkandthepart, v h t + v t .Thesecond periodisalsoastand-aloneperiodmimickingtherstperiodunderdifferentcost parametersanddemanddistributionforthepart. Inthesecondpolicy,whichwecallas stand-alonerstperiodwithpartcarry-over thedemanufacturer'srstperiodproblemisthesameasinthe stand-aloneperiods modelforthepurposeofoptimizingthisperiod'sexpectedtotalprots.However,instead ofsalvagingtherecoveredpartsremaininginstockattheendoftherstperiod,in reality,thedemanufacturerkeepsalloftheremainingstockofrecoveredpartsforthe second-perioddemand,andsellsanyremaininginventoryoftheend-of-lifeproducts tothematerialrecycler.Theinventorycarried-overfromtherstperiodisshownwith 85

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I R = I R 1 0 ,wherewedroptheindexcorrespondingtotheperiodfrom I R t )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 since weassumenoinitialinventoryatthebeginningoftherstperiodandthattheremight beinitialstockcarriedfromtherstperiodonly.Weusethetypeofstockandtherst periodindextodifferentiatebetweentheend-of-lifeproductsandtherecoveredparts. Inthethirdpolicy,whichwecallas stand-alonerstperiodwithpartrecoveryand carry-over ,thedemanufacturer'srstperiodexpectedtotalprotfunctionisoptimized assumingtherstperiodisastand-aloneperiod,similartothersttwomodels.In reality,allend-of-lifeproductsremaininginstockattheendoftherstperiodare demanufactured,andcarried-overintothenextperiodtogetherwiththeremaining recoveredpartsfromperiodone,i.e., I R = I R 1 + I P 1 ,where I R 1 I P 1 0 correspondto thequantitiesofrecoveredpartsandend-of-lifeproductsleft-overattheendoftherst periodrespectively.Inallmodels,weassumethattheremainingitemsattheendof periodtwocanbesoldtothematerialrecyclerfortheirmaterialcontent. Inventoryholdingcostsarenotconsideredintheoptimizationoftherstperiod problemsincethisperiodisassumedtobeastand-aloneperiod.However,onceboth periodproblemsareoptimized,theexpectedtotalprotfunctionismodiedtoinclude therelevantinventoryholdingcosts h peritemcarried-over,aswellastheactual operationalcostsandsalvagerevenuesdependingonthepolicy.Furthermore,wenote thatallcostparametersreecttheirnetpresentvalues.Ourdecisionvariablesinthis settingarethequantitiesofend-of-lifeproductsacquiredtoallocateamongthestocks pointsfor:end-of-lifeproducts P t ,andrecoveredparts R t ,whicharedetermined independentofthesecondperiodfortheinitialperiod,anddependingontheending inventoryoftherstperiodinthelastperiod. 5.3ModelDevelopmentandAnalysis Welabelthe stand-aloneperiods modelas SAP stand-alonerstperiodwith partcarry-over modelas SAP-CR ,and stand-alonerstperiodwithpartrecoveryand carry-over modelas SAP-RCR .Wedescribeeachmodelinthefollowingsections. 86

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5.3.1SAP:Stand-alonePeriods Under SAP ,thedemanufacturerconsiderseachperiodasastand-aloneperiod, andtherefore,discardsanyremainingstockofend-of-lifeproductsandrecoveredparts. Inthiscase,bothperiodshavethesamefunctionalformfortheexpectedtotalprots, butwithdifferentcosttermsanddemanddistributions,expressedasfollowsfor t =1,2 : SAP t : SAP t R t P t = Z y t R t 0 p t x t f t x t dx t | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1. a + Z y t R t + P t y t R t )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(p t y t R t + p t )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r t x t )]TJ/F39 11.9552 Tf 11.955 0 Td [(y t R t f t x t dx t | {z } 5 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1. b + Z 1 y t R t + P t )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(p t y t R t + p t )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r t y t P t f t x t dx t | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1. c + Z y t R t + P t 0 v t )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(y t R t + P t )]TJ/F39 11.9552 Tf 11.955 0 Td [(x t f t x t dx t | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1. d )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z 1 y t R t + P t r t )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(x t )]TJ/F39 11.9552 Tf 11.956 0 Td [(y t R t + P t f t x t dx t | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1. e )]TJ/F22 11.9552 Tf 11.291 0 Td [( c t + c i t )]TJ/F39 11.9552 Tf 11.955 0 Td [(v h t R t + P t | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1. f )]TJ/F39 11.9552 Tf 11.291 0 Td [(h p t y t P t | {z } 5 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1. g )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r t y t R t | {z } 5 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1. h + v t )]TJ/F39 11.9552 Tf 11.955 0 Td [(y t R t + P t | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1. i -1 Thetermsin5-1.a,5-1.b,and5-1.crepresentthegrossprotsearnedfromthe salesminusthecostofdemanufacturingoperationsafterobservingthedemand,and aregivenbasedontheavailablequantityoftherecoveredpartsandend-of-lifeproducts instock.Salvageisearnedfromtheremainingstockattheendoftheperiodincase thedemandislessthanthestockquantity,expressedinterm5-1.d.Ifthedemand exceedstheavailablequantityofitems,thedemanufacturerlosestheunmetportion ofthedemand,andincurstherespectiveloss-of-goodwillcostgiveninterm5-1.e. Thetermsin5-1.fcorrespondtotheacquisitionandtestingcostsincurred,andthe salvagerevenuesearnedfromthehulkofacquiredend-of-lifeproducts.Environmental feesareincurredfortheend-of-lifeproductskeptinstockgiveninterm5-1.g,and theoperationalcostsareincurredfortherecoveredpartsgiveninterm5-1.h.Finally, thetermin5-1.irepresentsthepartsalvagerevenuesobtainedfromtheend-of-life productsthatfailtopassthediagnostictestduetounrecoverablepartsonthem. 87

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Expectedtotalprotsoverbothperiodsisthesumoftheexpectedprotsineach individualperiod,i.e., SAP R 1 R 2 P 1 P 2 = SAP 1 R 1 P 1 + SAP 2 R 2 P 2 .Theexpected protinequation5-1isstrictlyconcaveinthedecisionvariables, R t P t ,asstatedin Theorem5.1: Theorem5.1. SAP t R t P t isstrictlyconcavein R t and P t for t =1,2 ifandonly ifthefollowingconditionshold:i p t )]TJ/F39 11.9552 Tf 12.55 0 Td [(c r t > v t )]TJ/F39 11.9552 Tf 12.55 0 Td [(r t ,i.e.,satisfyingthereusablepart demandfromthestockofend-of-lifeproductsismoreprotablethansalvagingthepart, earningthesalvagerevenueandincurringthepenaltyforthelostsale,ii c r t > 0 ,i.e., themarginalcostofpostponingtherecoveryoperationispositive. Proof. FirstorderoptimalityconditionsandtheHessianmatrix, H ,aregiven 8 t =1,2 as: SAP 0 R t = )]TJ/F39 11.9552 Tf 11.956 0 Td [(y t p t )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r t )]TJ/F39 11.9552 Tf 11.955 0 Td [(v t + r t F t y t R t + P t )]TJ/F39 11.9552 Tf 11.955 0 Td [(y t c r t F t y t R t + y t p t )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r t )]TJ/F39 11.9552 Tf 11.955 0 Td [(v t + r t )]TJ/F39 11.9552 Tf 11.955 0 Td [(c t )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i t + v t + v h t SAP 0 P t = )]TJ/F39 11.9552 Tf 11.956 0 Td [(y t p t )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r t )]TJ/F39 11.9552 Tf 11.955 0 Td [(v t + r t F t y t R t + P t + y t p t )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r t )]TJ/F39 11.9552 Tf 11.955 0 Td [(v t + r t )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p t )]TJ/F39 11.9552 Tf 11.955 0 Td [(c t )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i t + v t + v h t H = )]TJ/F28 7.9701 Tf 7.472 -4.783 Td [()]TJ/F23 7.9701 Tf 6.587 0 Td [( A + B )]TJ/F40 7.9701 Tf 6.586 0 Td [(A )]TJ/F40 7.9701 Tf 6.587 0 Td [(A )]TJ/F40 7.9701 Tf 6.586 0 Td [(A with A = y 2 t p t )]TJ/F39 11.9552 Tf 12.565 0 Td [(c r t )]TJ/F39 11.9552 Tf 12.564 0 Td [(v t + r t f t y t R t + P t ,and B = y 2 t c r t f t y t R t H isnegative deniteifandonlyifthethetermsinparenthesesinsidetheintegralsfor A and B are positive.Wefurthernotethatconditioniiisunnecessaryunderthenon-trivialcostof disassemblyplusrecovery,i.e.,theoperationalcosts,andismerelypresentedforthe sakeofcompleteness. Similartothesolutionapproachusedsofarinthisdissertation,wecansolvethe rstorderconditionsfortheoptimalvaluesofthedecisionvariablesbasedontheKKT conditionsineachperiod.However,duetosimplicityoftheprotfunction,thesolutionis simplertoobtain.Sincetherearetwodecisionvariablesinagivenperiod t =1,2 ,there arefourpossibleKKTcases,whichweoutlineinTable5-1below: 88

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Table5-1.SAP:KKTcases. Case R t P t SAP 0 R t SAP 0 P t Explanation C1 0 0 00Bothstockpointscanbeutilized. C2 0 00 < 0 Onlystockpointfortherecoveredpartscanbeutilized. C30 0 < 0 0Onlystockpointfortheend-of-lifeproductscanbeutilized. C400 < 0 < 0 Donotoperatethedemanufacturingbusiness. C1isthestationarypointcorrespondingtoutilizingbothstockpoints,andisoptimal ifcertainconditionsaresatised.C2correspondstostockingonlytherecoveredparts. C3iseliminatedsinceitisnotfeasible.Underpositiveenvironmentalfees,thesolution obtainedfromtherstorderconditioncorrespondingtotheend-of-lifeproductsdoes notsatisfytherequirementfortherstorderconditioncorrespondingtotherecovered partstobenegativeunderpositiveenvironmentalfees.Undertheabsenceofthe environmentalfees,C3iscoveredbyC1inwhichonlyend-of-lifeproductsarekeptin stock.Finally,C4correspondstothecaseinwhichitisnotprotabletooperatethe demanufacturingbusiness. InTheorem5.2,wegivetheconditionsunderwhicheachKKTcaseholdsand showthattheformoftheoptimalpolicyineachperiodisverysimpletoobtain.Under concavity,thedemanufacturingyieldratedeterminesifthedemanufacturingbusinessis protableornot.Ifitisprotabletooperatethedemanufacturingbusiness,thenwhich KKTcaseholdsisdeterminedbycomparingthreethresholdvalues.Thesethreshold valuesarethecriticalvaluesobtainedfromprobabilitiesofnotstockingout,anddepend ontherelativemagnitudesoftheenvironmentalfeesandtheoperationalcosts,i.e.,the trade-offbetweentheenvironmentalfeesandtheoperationalcosts. Theorem5.2. Let y LB SAP t = c t + c i t )]TJ/F39 11.9552 Tf 11.571 0 Td [(v t )]TJ/F39 11.9552 Tf 11.571 0 Td [(v h t = p t )]TJ/F39 11.9552 Tf 11.571 0 Td [(c r t )]TJ/F39 11.9552 Tf 11.571 0 Td [(v t + r t forperiod t =1,2 Ifthedemanufacturingyieldrate, y t y LB SAP t ,thenitisnotprotabletodemanufacturein thatperiod.Otherwise,let 1, t = h p t = c r t 2, t = )]TJ/F39 11.9552 Tf 5.479 -9.683 Td [(y t p t )]TJ/F39 11.9552 Tf 10.032 0 Td [(c r t )]TJ/F39 11.9552 Tf 10.033 0 Td [(v t + r t )]TJ/F39 11.9552 Tf 10.033 0 Td [(h p t )]TJ/F39 11.9552 Tf 10.033 0 Td [(c t )]TJ/F39 11.9552 Tf 10.033 0 Td [(c i t + v t + v h t = y t p t )]TJ/F39 11.9552 Tf 9.828 0 Td [(c r t )]TJ/F39 11.9552 Tf 9.827 0 Td [(v t + r t ,and 3, t = )]TJ/F39 11.9552 Tf 5.48 -9.683 Td [(y t p t )]TJ/F39 11.9552 Tf 9.827 0 Td [(c r t )]TJ/F39 11.9552 Tf 9.828 0 Td [(v t + r t )]TJ/F39 11.9552 Tf 9.828 0 Td [(c t )]TJ/F39 11.9552 Tf 9.828 0 Td [(c i t + v t + v h t = y t p t )]TJ/F39 11.9552 Tf 9.828 0 Td [(v t + r t Then,theoptimalstockingpolicy y t R t y t P t andhencetheoptimalacquisition R t P t policyisdescribedasinTable5-2: 89

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Table5-2.SAP:Optimalacquisitionandstockingpolicy. ConditionsonprobabilityAllocationQuantities Trade-offconditionofnotstockingoutKKTCase y t R t y t P t h p t =0 C10 F )]TJ/F23 6.9738 Tf 6.226 0 Td [(1 t 2, t 0 < h p t < c r t 1, t < 2, t C1 F )]TJ/F23 6.9738 Tf 6.226 0 Td [(1 t 1, t F )]TJ/F23 6.9738 Tf 6.227 0 Td [(1 t 2, t )]TJ/F39 9.9626 Tf 9.962 0 Td [(F )]TJ/F23 6.9738 Tf 6.227 0 Td [(1 t 1, t 1, t = 2, t C1 F )]TJ/F23 6.9738 Tf 6.226 0 Td [(1 t 1, t 0 1, t > 2, t C2 F )]TJ/F23 6.9738 Tf 6.226 0 Td [(1 t 3, t 0 h p t c r t C2 F )]TJ/F23 6.9738 Tf 6.226 0 Td [(1 t 3, t 0 Proof. SolvingC1yieldstheprobabilitiesofnotstockingout: F t y t R t = 1, t ,and F t y t R t + P t = 2, t ,whichmustsatisfy 0 1, t 2, t 1 andnotboth 1, t and 2, t canbeone.Similarly,solvingC2yields: F t y t R t = 3, t ,whichmustsatisfy 3, t 2 ,1 ,and 2, t < 3, t < 1, t thatcanbeobtainedfromthepartialderivativeof theobjectivefunctionwithrespectto P t .Then,wehavethefollowingcases: 1.When h p t =0 1, t =0 bydenition,and 2, t 2 ,1 issatisedsincethe demanufacturingbusinessisassumedtobeprotable,resultinginthestationary pointgivenbyC1tobetheoptimalsolutionwithonlyend-of-life-productsstock pointcontainingitems. 2.When 0 < h p t < c r t 1, t 2 ,1 bydenition.Inthiscase: aIf 1, t < 2, t ,then 2, t < 1 duetoprotabilityofthebusiness,andhence thestationarypointgivenbyC1isoptimalwithpositiveallocationquantitiesin bothstockpoints, bIf 1, t = 2, t ,then 2, t = 3, t holdsbytheformerequality.Inthiscase,C2 doesnothold,whereasC1holdswithonlyrecoveredpartsinstock, cIf 1, t > 2, t ,thenC1doesnotholdwhereasC2holdssince 3, t 2 ,1 due toprotabilityofthebusiness,and 2, t < 3, t < 1, t holds.Therefore,itis optimaltokeeponlyrecoveredpartsinstock. 3.When h p t c r t 1, t 1 bydenition,and 2, t < 1 ,resultingin 1, t > 2, t Therefore,C1doesnothold.Furthermore, 1, t > 3, t issatisedsince 1, t 1 and 3, t 2 ,1 duetoprotabilityofthedemanufacturingbusiness.Furthermore, 2, t < 3, t holds,resultinginC3tobetheoptimalKKTcasewhereonlyrecovered partsarekeptinstock.Theproofof 2, t < 3, t isbycontradiction:Assume 2, t 3, t holds.Let A t = y t p t )]TJ/F39 11.9552 Tf 13.169 0 Td [(c r t )]TJ/F39 11.9552 Tf 13.169 0 Td [(v t + r t )]TJ/F39 11.9552 Tf 13.169 0 Td [(c t )]TJ/F39 11.9552 Tf 13.169 0 Td [(c i t + v t + v h t B t = y t p t )]TJ/F39 11.9552 Tf 12.237 0 Td [(v t + r t .Then,by 2, t 3, t A )]TJ/F39 11.9552 Tf 12.237 0 Td [(y t h p t = B )]TJ/F39 11.9552 Tf 12.237 0 Td [(y t c r t A = B ,from which A t = B t h p t = c r t ,orequivalently, 3, t 1, t ,whichcontradictsthefactthat 1, t > 3, t 90

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Theorem5.2statesthatifthedemanufacturingyieldrateisatorbelowacertain threshold,thenitisoptimalnottodemanufacture.Ifthedemanufacturingyield rateexceedsthethreshold,thenwehavethefollowingcases:Intheabsenceof environmentalfeesoriftheenvironmentalfeesarenegligiblecomparedtooperational costs,thenitisoptimaltoacquireandkeeponlyend-of-lifeproducts.If,ontheother hand,environmentalfeesareclosetotheoperationalcostsinmagnitude,thenitis optimaltoacquireend-of-lifeproducts,demanufacturesomeportionofthem,andkeep bothend-of-lifeproductsandrecoveredpartsinstock.Finally,iftheenvironmentalfees arehigherthantheoperationalcosts,thenitisoptimaltoacquireend-of-lifeproducts, demanufacturethem,andkeeponlyrecoveredparts.Thethresholdvalues, i t for i =1,2,3 ,and t =1,2 ,denetherangeswherethetrade-offbetweentheenvironmental feesandoperationalcostsdetermineswhichstockingorallocationpolicyisoptimal. 5.3.2SAP-CR:Stand-aloneFirstPeriodwithPartCarry-over Under SAP-CR ,therstperiodproblemisthesameastheoneunder SAP .Inother words,thedemanufactureroptimizeshisacquisitionandallocationquantitiesinthe rstperiodassuminghelosestheunmetdemandandsalvagesanyleft-overstockof end-of-lifeproductsandrecoveredpartsattheendoftherstperiod.Atthebeginning ofthesecondperiod,thedemanufacturerisassumedtoobtaininformationaboutthe newgenerationoftheproduct,andthereusablepartonit.Notethatifthepartonthe newproducthasadrasticdesignchangefromtheoneontheoldergenerationofthe product,thenthedemanufacturermayuse SAP ,wherehesalvagesallleft-overitemsat theendoftherstperiod. Assumingthereispartcommonalitybetweentheoldandthenewgenerations oftheproduct,insteadofsalvagingtherecoveredpartsleftinstockattheendofthe rstperiod,thedemanufacturercarriesthemovertothesecondperiod,andsalvages onlytheremainingperiod-onestockofend-of-lifeproducts.Therefore,theinitialstock ofrecoveredpartsisknownbythedemanufacturer,andhehastodecideonthe 91

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acquisitionandallocationquantitiesinthesecondperiod,i.e., R 2 and P 2 .Theactual expectedprotfunctionintherstperiodisobtainedbymodifyingtheoneunder SAP accordingtothepolicystructure after theoptimizationiscompleted,andtheoptimal valuesoftheacquisitionquantitiesforthestand-aloneperiodareobtained. Lettheoptimalacquisitionquantitiesofend-of-lifeproductstokeepinthestock pointsofrecoveredparts,andend-of-lifeproductsbedenotedby R SAP 1 ,and P SAP 1 respectively.Theactualexpectedprotinperiodonecanbewrittenasinequation5-2: SAP-CR 1 : SAP )]TJ/F40 7.9701 Tf 6.586 0 Td [(CR 1 R SAP 1 P SAP 1 = Z y 1 R SAP 1 0 )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(p 1 x 1 )]TJ/F39 11.9552 Tf 11.956 0 Td [(h y 1 R SAP SAP 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(x 1 f 1 x 1 dx 1 | {z } 5 )]TJ/F23 7.9701 Tf 6.586 0 Td [(2. a + Z y 1 R SAP 1 + P SAP 1 y 1 R SAP 1 )]TJ/F39 11.9552 Tf 5.48 -9.683 Td [(p 1 y 1 R SAP 1 + p 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r ,1 x 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(y 1 R SAP 1 f 1 x 1 dx 1 | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2. b + Z 1 y 1 R SAP 1 + P SAP 1 )]TJ/F39 11.9552 Tf 5.479 -9.683 Td [(p 1 y 1 R SAP 1 + p 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r ,1 y 1 P SAP 1 f 1 x 1 dx 1 | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2. c + Z y 1 R SAP 1 + P SAP 1 y 1 R SAP 1 v 1 )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(y 1 R SAP 1 + P SAP 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(x 1 f 1 x 1 dx 1 | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2. d )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z 1 y 1 R SAP 1 + P SAP 1 r 1 )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(x 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(y 1 R SAP 1 + P SAP 1 f 1 x 1 dx 1 | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2. e )]TJ/F22 11.9552 Tf 11.291 0 Td [( c 1 + c i ,1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(v h ,1 R SAP 1 + P SAP 1 | {z } 5 )]TJ/F23 7.9701 Tf 6.586 0 Td [(2. f )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p ,1 y 1 P SAP 1 | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2. g )]TJ/F39 11.9552 Tf 11.291 0 Td [(c r ,1 y 1 R SAP 1 | {z } 5 )]TJ/F23 7.9701 Tf 6.586 0 Td [(2. h + v 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(y 1 R SAP 1 + P SAP 1 | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(2. i -2 Thetermsin5-2.aand5-2.daremodiedfromthoseintheexpectedprot functionunder SAP .Theformertermincludestheinventoryholdingcostsincurredfor theremainingstockofrecoveredpartsthatarecarried-overtothenextperiod,andthe latterterm'sintegralboundisupdatedtoreectthefactthatsalvagevaluesfortheparts canonlybeearnedfromtheend-of-lifeproductsthatremaininstockattheendofperiod one.Givenaninitialquantityofrecoveredparts, I R 0 I R = I R 1 + I P 1 where I P 1 =0 ,the expectedtotalprotsinthesecondperiodcanbewrittenasinequation5-3: 92

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SAP-CR 2 : SAP )]TJ/F40 7.9701 Tf 6.586 0 Td [(CR 2 R 2 P 2 j IR = Z y 2 R 2 + I R 0 p 2 x 2 f 2 x 2 dx 2 + Z y 2 R 2 + P 2 + I R y 2 R 2 + I R p 2 y 2 R 2 + I R + p 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r ,2 x 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(y 2 R 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(I R f 2 x 2 dx 2 + Z 1 y 2 R 2 + P 2 + I R )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(p 2 y 2 R 2 + I R + p 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r ,2 y 2 P 2 f 2 x 2 dx 2 + Z y 2 R 2 + P 2 + I R 0 v 2 )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(y 2 R 2 + P 2 + I R )]TJ/F39 11.9552 Tf 11.955 0 Td [(x 2 f 2 x 2 dx 2 )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z 1 y 2 R 2 + P 2 + I R r 2 )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(x 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(y 2 R 2 + P 2 f 2 x 2 dx 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [( c 2 + c i ,2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(v h ,2 R 2 + P 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p ,2 y 2 P 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r ,2 y 2 R 2 + v 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(y 2 R 2 + P 2 -3 Noticethatequation5-3issimilartoequation5-1under SAP ,whereintegral boundsandtermsaremodiedbasedontheinitialquantityoftherecoveredparts. SAP )]TJ/F40 7.9701 Tf 6.587 0 Td [(CR 2 isstrictlyconcaveunderthesameconditionsgivenfor SAP 2 ,andtherefore, relateddiscussionwillnotberepeated.Firstorderconditionsfor SAP )]TJ/F40 7.9701 Tf 6.587 0 Td [(CR 2 R 2 P 2 are: SAP )]TJ/F40 7.9701 Tf 6.587 0 Td [(CR 2 0 R 2 = )]TJ/F39 11.9552 Tf 11.955 0 Td [(y 2 p 2 )]TJ/F39 11.9552 Tf 11.956 0 Td [(c r ,2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(v 2 + r 2 F 2 y 2 R 2 + P 2 + I R )]TJ/F39 11.9552 Tf 11.955 0 Td [(y 2 c r ,2 F 2 y 2 R 2 + I R + y 2 p 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r ,2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(v 2 + r 2 )]TJ/F39 11.9552 Tf 11.956 0 Td [(c 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i ,2 + v 2 + v h ,2 SAP )]TJ/F40 7.9701 Tf 6.586 0 Td [(CR 2 0 P 2 = )]TJ/F39 11.9552 Tf 11.955 0 Td [(y 2 p 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r ,2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(v 2 + r 2 F 2 y 2 R 2 + P 2 + I R + y 2 p 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r ,2 )]TJ/F39 11.9552 Tf 11.956 0 Td [(v 2 + r 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p ,2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c 2 )]TJ/F39 11.9552 Tf 11.956 0 Td [(c i ,2 + v 2 + v h ,2 Wedenethefollowingquantities: Q 1 = F )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 1,2 Q 2 = F )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 2,2 Q 3 = F )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 3,2 ,i.e.,thecriticalvalueswhichdeterminethesecond-period-acquisition quantitiesofend-of-lifeproductstoallocateamongthestockpoints.KKTcasesand theirsolutionsalongwiththenecessaryconditionsaregiveninCorollary5.2.1below: Corollary5.2.1. KKTcasesforSAP-CR 2 andtheirsolutionsaregivenasfollows: [C1] End-of-lifeproductscanbeacquiredandallocatedamongbothstockpoints inthesecondperiodinadditiontotherecoveredpartsleft-overfromtherstperiod. R 2 P 2 0 SAP )]TJ/F40 7.9701 Tf 6.587 0 Td [(CR 2 0 R 2 = SAP )]TJ/F40 7.9701 Tf 6.586 0 Td [(CR 2 0 P 2 =0 F 2 y 2 R 2 + I R = h p ,2 c r ,2 = 1,2 F 2 y 2 R 2 + P 2 + I R = y 2 p 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r ,2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(v 2 + r 2 )]TJ/F39 11.9552 Tf 11.956 0 Td [(h p ,2 )]TJ/F39 11.9552 Tf 11.956 0 Td [(c 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i ,2 + v 2 + v h ,2 y 2 p 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r ,2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(v 2 + r 2 = 2,2 93

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fromwhich, y 2 R 2 = Q 1 )]TJ/F39 11.9552 Tf 12.302 0 Td [(I R = Q 1 )]TJ/F39 11.9552 Tf 12.302 0 Td [(I R 1 ,and y 2 P 2 = Q 2 )]TJ/F39 11.9552 Tf 12.302 0 Td [(Q 1 .Since I R y 2 R 2 y 2 P 2 0 0 I R Q 1 Q 2 mustbesatisedalongwith 0 1,2 2,2 .Notethat 2,2 < 1 holds. [C2] End-of-lifeproductscanbeacquiredandallocatedtothestockpointforthe recoveredpartsinthesecondperiodinadditiontotherecoveredpartsleft-overfromthe rstperiod. R 2 0 P 2 =0 SAP )]TJ/F40 7.9701 Tf 6.587 0 Td [(CR 2 0 R 2 =0, SAP )]TJ/F40 7.9701 Tf 6.586 0 Td [(CR 2 0 P 2 < 0 F 2 y 2 R 2 + I R = y 2 p 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r ,2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(v 2 + r 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i ,2 + v 2 + v h ,2 y 2 p 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(v 2 + r 2 = 3,2 fromwhich, y 2 R 2 = Q 3 )]TJ/F39 11.9552 Tf 12.246 0 Td [(I R = Q 3 )]TJ/F39 11.9552 Tf 12.246 0 Td [(I R 1 .Since I R y 2 R 2 0 ,and SAP )]TJ/F40 7.9701 Tf 6.586 0 Td [(CR 2 0 P 2 < 0 must hold, 0 I R Q 3 < Q 1 ,and Q 2 < Q 3 mustbesatised.Notethat 0 < 3,2 < 1 holds. [C3] End-of-lifeproductscanbeacquiredandallocatedtothestockpointforthe end-of-lifeproductsinthesecondperiodinadditiontotherecoveredpartsleft-overfrom therstperiod. R 2 =0 P 2 0 SAP )]TJ/F40 7.9701 Tf 6.587 0 Td [(CR 2 0 R 2 < 0, SAP )]TJ/F40 7.9701 Tf 6.586 0 Td [(CR 2 0 P 2 =0 F 2 y 2 P 2 + I R = y 2 p 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r ,2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(v 2 + r 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p ,2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i ,2 + v 2 + v h ,2 y 2 p 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r ,2 )]TJ/F39 11.9552 Tf 11.956 0 Td [(v 2 + r 2 = 2,2 fromwhich, y 2 P 2 = Q 2 )]TJ/F39 11.9552 Tf 12.249 0 Td [(I R = Q 2 )]TJ/F39 11.9552 Tf 12.249 0 Td [(I R 1 .Since I R y 2 P 2 0 ,and SAP )]TJ/F40 7.9701 Tf 6.587 0 Td [(CR 2 0 R 2 < 0 must hold, Q 1 < I R Q 2 mustbesatised.Notethat 0 < 2,2 < 1 holds. [C4] Donotacquireanyadditionalend-of-lifeproductsinthesecondperiod. R 2 P 2 =0 SAP )]TJ/F40 7.9701 Tf 6.586 0 Td [(CR 2 0 R 2 < 0 SAP )]TJ/F40 7.9701 Tf 6.587 0 Td [(CR 2 0 P 2 < 0 .Since I R 0 SAP )]TJ/F40 7.9701 Tf 6.586 0 Td [(CR 2 0 R 2 < 0 ,and SAP )]TJ/F40 7.9701 Tf 6.586 0 Td [(CR 2 0 P 2 < 0 musthold, I R > Q 3 and I R > Q 2 mustholdaswell.Dependingonthe relativevaluesof Q 1 and Q 2 ,wehavetwosub-cases: [C4.1] When Q 3 < Q 2 Q 3 < Q 2 < I R musthold,and [C4.2] When Q 3 Q 2 Q 2 Q 3 < I R musthold, whichareobtainedfromtherstorderconditions. C1correspondstothecasewhentheleft-overinventoryofrecoveredpartsfrom therstperiodmaynotbesufcienttomeetthedemandinthesecondperiod,hence end-of-lifeproductscanbeacquiredandallocatedamongstockpointsdependingon thevalueof I R andthecriticalvalues.C2correspondstothecasewhentheleft-over 94

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inventoryofrecoveredpartsfromtherstperiodmayormaynotbesufcienttomeet thedemand,inwhichcaseend-of-lifeproductsmaybeacquiredinadditiontothe left-overstockoftherecoveredparts,andtheenvironmentalfeesmaybehighenough tonecessitateprioritizingthedemanufacturingoperations,andkeepingonlyrecovered partsinstock.C3correspondstothecasewhentheleft-overinventoryofrecovered partsfromtherstperiodmayormaynotbesufcienttomeetthedemand,inwhich caseend-of-lifeproductsmaybeacquiredinadditiontotheleft-overstockofthe recoveredparts,andtheenvironmentalfeesarelowenoughleadingtopostponing thedemanufacturingoperations,andkeepingtheacquireditemsinthestockpoint ofend-of-lifeproducts.C4correspondstothecasewhenthequantityoftheleft-over inventoryofrecoveredpartsfromtherstperiodissufcienttomeetthedemand,in whichcasenoadditionalend-of-lifeproductsareneededtobeacquired. WhichKKTcaseholdsdependsontheleft-overquantityfromtherstperiod, I R andthetrade-offbetweentheenvironmentalfeesandtheoperationalcostswhichaffect therelativerankingofthecriticalvalues, Q 1 Q 2 Q 3 .InTheorem5.3below,weshow therankingofthesecriticalvaluescantakeonlyspecicformsbasedonthetrade-off betweentheenvironmentalfeesandtheoperationalcosts: Theorem5.3. Criticialvalues, Q 1 Q 2 Q 3 ,takeexactlyoneofthefollowingrankings: 1.If h p ,2 =0 ,then: aif Q 1 < Q 2 ,then 0= Q 1 < Q 3 < Q 2 holds, bif Q 1 = Q 2 ,then Q 1 = Q 2 = Q 3 =0 holds,and cif Q 1 > Q 2 ,then Q 2 < Q 3 < Q 1 =0 holds. 2.If 0 < h p ,2 < c r ,2 ,then Q 1 > 0 holdsbydenition,and: aif Q 1 < Q 2 ,then 0 < Q 1 Q 3 < Q 2 holds, bif Q 1 = Q 2 ,then 0 < Q 1 = Q 2 = Q 3 holds,and cif Q 1 > Q 2 ,then Q 2 < Q 3 < Q 1 holds. 3.If h p ,2 c r ,2 ,then Q 1 1 and Q 2 < Q 3 < Q 1 holds. 95

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Proof. Let A = y 2 p 2 )]TJ/F39 11.9552 Tf 10.881 0 Td [(c r ,2 )]TJ/F39 11.9552 Tf 10.881 0 Td [(v 2 + r 2 B = y 2 p 2 )]TJ/F39 11.9552 Tf 10.882 0 Td [(v 2 + r 2 ,whicharebothnonnegative duetoconcavitycondition.Hence, Q 2 = A )]TJ/F39 11.9552 Tf 11.955 0 Td [(y 2 h p ,2 = B )]TJ/F39 11.9552 Tf 11.956 0 Td [(y 2 c r ,2 ,and Q 3 = A = B 1.When h p ,2 =0 Q 1 =0 bydenition. aIf Q 1 < Q 2 ,then 0 < Q 2 = A = B )]TJ/F39 11.9552 Tf 12.621 0 Td [(y 2 c r ,2 ,fromwhich A > 0 ,andhence Q 3 = A = B > Q 1 =0 musthold.Proofof Q 3 < Q 2 isbycontradiction:Assume, onthecontrary,that Q 3 Q 2 ,i.e, A = B A = B )]TJ/F39 11.9552 Tf 12.281 0 Td [(y 2 c r ,2 holds.Rearranging thisinequalityyields Ay 2 c r ,2 0 ,whichcontradictsthefactthat A > 0 bIf Q 1 = Q 2 ,then A =0 musthold,asaresultofwhich Q 3 = A = B =0 mustbe satisedunderconcavity. cIf Q 1 > Q 2 ,then A < 0 musthold,fromwhich Q 3 = A = b > A = B )]TJ/F39 11.9552 Tf 10.908 0 Td [(y 2 c r ,2 = Q 2 2.When 0 < h p ,2 < c r ,2 0 < Q 1 < 1 holdsbydenition. aIf Q 1 < Q 2 ,then A )]TJ/F39 11.9552 Tf 12.627 0 Td [(y 2 h p ,2 > 0 ,and h p ,2 = c r ,2 < A )]TJ/F39 11.9552 Tf 12.628 0 Td [(y 2 h p ,2 = B )]TJ/F39 11.9552 Tf 12.628 0 Td [(y 2 c r ,2 holds.Rearrangingthesecondinequalityyields A = B > h p ,2 = c r ,2 ,i.e., Q 3 > Q 1 Proofof Q 3 < Q 2 isbycontradiction:Assume,onthecontrary,that Q 3 Q 2 i.e., A = B A )]TJ/F39 11.9552 Tf 12.167 0 Td [(y 2 h p ,2 = B )]TJ/F39 11.9552 Tf 12.167 0 Td [(y 2 c r ,2 holds.Rearrangingthisinequalityyields A = B h p ,2 = c r ,2 ,i.e., Q 3 Q 1 whichiscontradictorytothecondition, Q 1 < Q 2 ofthiscase. bif Q 1 = Q 2 ,i.e., h p ,2 = c r ,2 = A )]TJ/F39 11.9552 Tf 12.551 0 Td [(y 2 h p ,2 = B )]TJ/F39 11.9552 Tf 12.551 0 Td [(y 2 c r ,2 ,thenrearrangingthis equalityyields A = B = h p ,2 = c r ,2 ,i.e., Q 3 = Q 1 musthold. cIf Q 1 > Q 2 ,then Q 3 < Q 1 holdsfroma.Proofof Q 2 < Q 3 isbycontradiction. Assume,onthecontrary,that Q 2 Q 3 ,i.e., A )]TJ/F39 11.9552 Tf 10.246 0 Td [(y 2 h p ,2 = B )]TJ/F39 11.9552 Tf 10.246 0 Td [(y 2 c r ,2 h p ,2 = c r ,2 Rearrangingthisinequalityyields A = B h p ,2 = c r ,2 ,i.e., Q 3 Q 1 ,whichwhich iscontradictorytothecondition, Q 1 > Q 2 ofthiscase. 3.When h p ,2 c r ,2 Q 1 1 bydenition.Therefore, Q 1 > Q 2 and Q 1 > Q 3 musthold underconcavity.Proofof Q 2 < Q 3 isbycontradiction:Assume,onthecontrary, that Q 2 Q 3 .Then,bytheproofinthelastbulletofii, Q 3 Q 1 =1 ,which contradictstherstpartofthisproof,i.e., Q 3 < Q 1 TherankingsinTheorem5.3canbematchedwiththeconditionsofeachKKTcase denedinCorollary5.2.1,fromwhichthesolutiontothesecondperiodmodelcanbe obtainedbasedonthetrade-offbetweentheenvironmentalfeesandtheoperational costs,andtheinitialstockquantityoftherecoveredparts.Figure5-2showswhichKKT casesholdunderpossiblerankingsofthecriticalvalues,andthetrade-offbetweenthe environmentalfeesandoperationalcosts: 96

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A B Figure5-2.Optimalsolutionof CAP-CR 2 basedoninitialinventorylevelandcritical thresholdvalues.A 0 h p ,2 < c r ,2 and Q 1 Q 2 .B 0 h p ,2 < c r ,2 and Q 1 > Q 2 ,or h p ,2 c r ,2 Examiningthesolution,onecanseethatabase-stockpolicyisoptimalwith order-up-tolevelof Q 2 inFigure5-2Aand Q 3 inFigure5-2B.Thebase-stockpolicy matcheswiththespecialstockingstructureweconsiderdependingonthetrade-off betweentheenvironmentalfeesandtheoperationalcosts.Forinstance,whenthe environmentalfeesaresmallerthantheoperationalcosts,iftheinitialinventorylevel isbelowtheorder-up-tolevel, Q 2 ,thenitisoptimaltoordermoreend-of-lifeproducts tobringituptothislevel.Thisorder-up-tolevelcanbebrokendownintoanother order-up-tolevelfortheacquisitionquantityofend-of-lifeproductstoallocatetothe stockpointfortherecoveredparts,givenby Q 1 Q 2 andwhentheinitialstock issmallerthanthislevel.Astheinitialstocklevelgoesabove Q 1 ,thepolicycan takeadvantageofthelowerenvironmentalfeestoacquirelessend-of-lifeproducts, eliminatingtheoperationalcostsastheprobabilitiesofnotstockingoutwillbesatised withalessadditionalacquisitionquantity,andinalessexpensiveway. Oncetheoptimalsolutionisobtainedforthesecondperiod,theactualexpected totalprotsunder SAP-CR canbefoundbyaddingthemodiedperiodoneexpected prot, SAP )]TJ/F40 7.9701 Tf 6.586 0 Td [(CR 1 R SAP 1 P SAP 1 inequation5-2,totheexpectedprotobtainedinthe secondperiod, SAP )]TJ/F40 7.9701 Tf 6.586 0 Td [(CR 2 R 2 P 2 j IR inequation5-3,where I R = R y 1 R SAP 1 0 y 1 R SAP 1 )]TJ/F39 11.9552 Tf -438.338 -23.908 Td [(x 1 f 1 x 1 dx 1 97

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5.3.3SAP-RCR:Stand-aloneFirstPeriodwithPartRecoveryandCarry-over Under SAP-RCR ,therstperiodproblemisthesameastheoneintheprevious models.Inotherwords,thedemanufactureroptimizeshisacquisitionandallocation quantitiesintherstperiodassuminghelosestheunmetdemand,salvagesany left-overinventoryofend-of-lifeproductsandrecoveredpartsattheendoftherst period.Atthebeginningofthesecondperiod,thedemanufacturerobtainsinformation aboutthenewgenerationoftheproductandthereusablepartonit.Assuming thereispartcommonalitybetweentheoldandthenewgenerationsoftheproduct, insteadofsalvagingtherecoveredpartsleftinstockattheendoftherstperiod,the demanufacturercarriesthemovertothesecondperiodalongwithanyremaining end-of-lifeproducts after performingthedemanufacturingoperations.Thehulk remainingaftertheend-of-lifeproductsaredemanufacturedaresoldtothematerial recycler.Thedemanufacturerhastodecideontheacquisitionandallocationquantities R 2 P 2 inthesecondperiodbasedontheinitialstockquantityoftherecoveredparts. Similarto SAP-CR 1 ,theactualexpectedprotfunctionintherstperiodisobtained bymodifyingtheoneunder SAP accordingtothepolicystructure after theoptimization iscompleted,andtheoptimalvaluesoftheacquisitionquantitiesforthestand-alone periodareobtained.Accordingly,theactualexpectedprotfunctionintherstperiod under SAP-RCR 1 canbewrittenasinequation5-4: SAP-RCR 1 : SAP )]TJ/F40 7.9701 Tf 6.587 0 Td [(RCR 1 R SAP 1 P SAP 1 = Z y 1 R SAP 1 + P SAP 1 0 p 1 x 1 f 1 x 1 dx 1 | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(4. a + Z 1 y 1 R SAP 1 + P SAP 1 p 1 y 1 R SAP 1 + P SAP 1 f 1 x 1 dx 1 | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(4. b )]TJ/F30 11.9552 Tf 11.291 16.272 Td [(Z y 1 R SAP 1 + P SAP 1 0 h )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(y 1 R SAP 1 + P SAP 1 )]TJ/F39 11.9552 Tf 11.956 0 Td [(x 1 f 1 x 1 dx 1 | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(4. c )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z 1 y 1 R SAP 1 + P SAP 1 r 1 )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(x 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(y 1 R SAP 1 + P SAP 1 f 1 x 1 dx 1 | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(4. d )]TJ/F22 11.9552 Tf 11.291 0 Td [( c 1 + c i ,1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(v h ,1 R SAP 1 + P SAP 1 | {z } 5 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4. e )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p ,1 y 1 P SAP 1 | {z } 5 )]TJ/F23 7.9701 Tf 6.586 0 Td [(4. f )]TJ/F39 11.9552 Tf 11.291 0 Td [(c r ,1 y 1 R SAP 1 + P SAP 1 | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(4. g + v 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(y 1 R SAP 1 + P SAP 1 | {z } 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(4. h -4 98

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Thetermsin5-4.aand5-4.bcorrespondtothesalesrevenues.Sinceall remaininginventoryofitemsarecarried-overtothenextperiodintheformofrecovered parts,operationalcostsareincurredforallacquiredend-of-lifeproductscontaining recoverablepartsgivenin5-4.g.Thedemanufacturerincursthecostofcarryingthe left-overinventoryofbothformsofitemsgivenin5-4.c.Furthermore,nosalvage revenuesareearnedfromthereusablepartsexceptfortheunrecoverableonesasgiven in5-4.h.Lost-salecostsin5-4.dareincurredfortheunmetportionofthedemand. Theremainingcosttermsaresimilartotheonesunder SAP-CR 2 Giventhequantityoftherecoveredpartscarried-over, I R = I R 1 + I P 1 I R 1 I P 1 0 ,the demanufacturer'sexpectedtotalprotfunctioninthesecondperiodisexactlythesame astheoneinequation5-3under SAP-CR 2 .Inotherwords,thecharacterizationofthe optimalsolutioninthesecondperiodunder SAP-RCR 2 isgivenbythatunder SAP-CR 2 wherethequantitiesofinterestmaytakedifferentoptimalvaluesduetopossiblyhigher initialstockatthebeginningofperiodtwounderthecurrentpolicy. Oncetheoptimalsecond-periodsolutionisobtained,theactualexpectedtotal protsunder SAP-RCR canbefoundbyaddingthemodiedperiodoneexpectedprot, SAP )]TJ/F40 7.9701 Tf 6.586 0 Td [(CR 1 R SAP 1 P SAP 1 inequation5-2,totheexpectedprotobtainedinthesecond periodinequation5-3,where I R = R y 1 R SAP 1 + P SAP 1 0 y 1 R SAP 1 + P SAP 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(x 1 f 1 x 1 dx 1 5.4ComputationalExperiments Inthissection,wenumericallycomparetheexpectedtotalprotsobtainedunder thethreemyopicpolicies, SAP SAP-RC and SAP-RCR ,andreportthesensitivityofthe expectedprotstoproblemparameters.Duetohighnumberofparametersinvolved,we xmostoftheparametersintherstperiod,andvarythevaluesofthesecond-period parameters.Furthermore,weassumethatallcostparametersinbothperiodsrepresent theirnetpresentvalues,sothatdiscountingisnotrequired. Wesettheunitacquisitioncost, c 1 ,at10,andvary c 2 atthreelevels,10,and15, allowingforcheaper,same,andmoreexpensiveacquisitioncostsinthesecondperiod. 99

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Similartoourpreviousnumericalstudies,wesetthesellingprice, p 1 ,ofthereusable partintherstperiodas50,andvarythesellingprice, p 2 ,inthesecondperiodatthree levels,50,and60.Thecostofthediagnostictest, c i t ,isxedat 0.5 inbothperiods. Inordertokeeptheexperimentsatamanageablelevel,thedemanufacturingyieldrate, Y t ,isvariedatfourlevels.80,0.85,0.90,0.95inbothperiods.Inordertocapture theimpactoftheenvironmentalfee, h p ,ontheacquisitionandallocationdecisions,we varyitatlow,medium,andhighlevels,takenasapercentageofthedemanufacturing cost,%,100%,and150%,alongwithacasewithnoenvironmentalfees.Totalcost ofthedisassemblyandrecoveryoperations, c r ,1 ,isxedat5fortherstperiod,and, c r ,2 isvariedatthreelevelsinthesecondperiod,7.5,10. Weuseuniformandnormaldistributionsforthedemand.Wexthemeandemand intherstperiod, 1 ,at30unitsperseason,andvary 2 inthesecondperiodatthree levels,30,60.Wexthestandarddeviation, t ,ofthedemandat10%ofthemean demandwhentheunderlyingdistributionisnormalinbothperiods.Theunitinventory holdingcost, h ,isxedat0.5;theunitloss-of-goodwillcost, r t ,istakenas20%ofthe sellingprice, p t ;andtheunitsalvagevalues, v t v h t ,aresetatxedvalues.5and 1.25inbothperiods, t =1,2 .Foragivendemanddistributionpairinperiods,wehave 20,736probleminstancesforeachmodel.Table5-3summarizesourparameterset: Table5-3.Parametersetforthemyopictwo-periodnewsvendormodels. p 1 p 2 c 1 c 2 c i ,1 c i ,2 y 1 y 2 h p ,1 h p ,2 c r ,1 c r ,2 h 1 2 1 2 r 1 r 2 v 1 v h ,1 v 2 v h ,2 50401050.50.800550.5301510202.5,1.25 50100.85507.530 60150.901001060 0.95150 %of: c r t t p t InFigures5-35-8,wepresenttheexpectedquantityoftherecoveredparts carried-overfromtherstperiodintothesecond,followedbythetotalacquisition quantityinthesecondperiod,andthentheexpectedtotalprotsobtainedunderrelevant modelsforaxeddemanufacturingyieldrateof0.90intherstperiod.Eachrowin 100

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thesegurescorrespondstoadifferenttrade-offscenario,i.e.,Anoenvironmentalfees, B50%,andC100%oftheoperationalcosts.InFigure5-3,allparametershavethe samevaluesexceptthedemanufacturingcostinthesecondperiodisdoubleitsvaluein therstperiod,andthedemandinbothperiodsareuniformlydistributed: A B C Figure5-3.Expectedquantitycarried-over,totalacquisitionquantityinthesecond period,andexpectedtotalprots, y 1 =0.90 c 1 c 2 =,10 c r ,1 c r ,2 =,10 X 1 U ,60 X 2 U ,60 p 1 p 2 =,50 A h p ,1 h p ,2 =,0 .B h p ,1 h p ,2 =0.5 c r ,1 c r ,2 C h p ,1 h p ,2 =1.5 c r ,1 c r ,2 SimilartoFigure5-3,Figure5-4showstheplotsundernormaldistributeddemand intherst,anduniformlydistributeddemandinthesecondperiodwiththesamemean: 101

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A B C Figure5-4.Expectedquantitycarried-over,totalacquisitionquantityinthesecond period,andexpectedtotalprots, y 1 =0.90 c 1 c 2 =,10 c r ,1 c r ,2 =,10 X 1 N ,3 X 2 U ,60 p 1 p 2 =,50 A h p ,1 h p ,2 =,0 .B h p ,1 h p ,2 =0.5 c r ,1 c r ,2 C h p ,1 h p ,2 =1.5 c r ,1 c r ,2 SAP-RCR outperforms SAP and SAP-CR inallexperiments.Therelativerankings of SAP and SAP-CR dependontheparametervalues,andinparticular,thedemand distributionsandthetrade-offbetweentheenvironmentalfeesandtheoperationalcosts. Evenwhentheacquisitionpriceinthesecondperiodislowerthantheoneintherst period,seeminglycreatinganadvantagetoacquireinthesecondperiod, SAP-RCR still resultsinthehighestexpectedtotalprotsseeFigures5-5and5-6.Thisisbecause 102

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ofthesmallvalueoftheholdingcostparameterusedinournumericalstudy.Verylittle inventoryholdingcostsincurredfortheitemscarried-overunder SAP-RCR offsetsthe marginsgeneratedbytheothermyopicpoliciesduetosmalleroperationalcostsin thesecondperiod,sincetheunitoperationalcostissignicantlyhigherthantheunit inventoryholdingcost.Inotherwords, SAP-RCR cantakeadvantageofbeingableto carry-overalargerquantityofitemsduetosmallinventoryholdingcosts. A B C Figure5-5.Expectedquantitycarried-over,totalacquisitionquantityinthesecond period,andexpectedtotalprots, y 1 =0.90 c 1 c 2 =,5 c r ,1 c r ,2 =,10 X 1 U ,60 X 2 U ,60 p 1 p 2 =,50 A h p ,1 h p ,2 =,0 .B h p ,1 h p ,2 =0.5 c r ,1 c r ,2 C h p ,1 h p ,2 =1.5 c r ,1 c r ,2 103

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A B C Figure5-6.Expectedquantitycarried-over,totalacquisitionquantityinthesecond period,andexpectedtotalprots, y 1 =0.90 c 1 c 2 =,5 c r ,1 c r ,2 =,10 X 1 N ,3 X 2 U ,60 p 1 p 2 =,50 A h p ,1 h p ,2 =,0 .B h p ,1 h p ,2 =0.5 c r ,1 c r ,2 C h p ,1 h p ,2 =1.5 c r ,1 c r ,2 Astheenvironmentalfeesincrease,thequantitycarried-overdecreasesslightly under SAP-RCR ,whereasitincreasesunder SAP-CR .Theformerisduetothefact thatoverallcostsincrease,soitismoreeconomicaltoacquireandstockless.The latterisbecause,thehighertheenvironmentalfeeswithrespecttooperationalcosts are,themoreeconomicalitbecomestoprioritizethedemanufacturingoperations,and tokeeprecoveredpartsasopposedtoend-of-lifeproductsinstock.Underpositive 104

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environmentalfees,theexpectedquantitycarried-overunder SAP-CR ishigherwhen therst-perioddemandisnormallydistributedascomparedtothequantitywhenthe rstperioddemandisuniformlydistributedseeFigures5-7and5-8corresponding touniformlydistributeddemands,andnormallydistributedrst-perioddemand followedbyauniformlydistributeddemand. SAP-CR outperforms SAP evenwhen theenvironmentalfeesaresmallerthantheoperationalcostsseeFigures5-3and5-4. A B C Figure5-7.Expectedquantitycarried-over,totalacquisitionquantityinthesecond period,andexpectedtotalprots, y 1 =0.90 c 1 c 2 =,5 c r ,1 c r ,2 =,5 X 1 U ,60 X 2 U ,60 p 1 p 2 =,50 A h p ,1 h p ,2 =,0 .B h p ,1 h p ,2 =0.5 c r ,1 c r ,2 C h p ,1 h p ,2 =1.5 c r ,1 c r ,2 105

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A B C Figure5-8.Expectedquantitycarried-over,totalacquisitionquantityinthesecond period,andexpectedtotalprots, y 1 =0.90 c 1 c 2 =,5 c r ,1 c r ,2 =,5 X 1 N ,3 X 2 U ,60 p 1 p 2 =,50 A h p ,1 h p ,2 =,0 .B h p ,1 h p ,2 =0.5 c r ,1 c r ,2 C h p ,1 h p ,2 =1.5 c r ,1 c r ,2 Therst-periodexpectedprotsarehigherunder SAP-CR ,sincenoadditional demanufacturingcostsareincurred,whereonlyalreadyrecoveredpartsarecarried-over tothenextperiod,andthesalvagerevenuesareearnedfromtheremainingend-of-life products.Furthermore,inventoryholdingcostsareincurredonlyfortherecoveredparts. Under SAP-RCR ,theremainingend-of-lifeproductshavetobeprocessedrst,resulting intheadditionaloperationalandholdingscosts,andnosalvagerevenues.However, 106

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SAP-RCR isabletoearnmoreprotsinthesecondperiodduetonotrequiringasmuch additionalacquisitionquantityinthesecondperiod,whereasothermyopicpolicies SAP-CR forbeingabletocarry-overlittleornone,and SAP fortreatingeachperiodas astand-aloneperiodhavetoacquiremoreadditionalend-of-lifeproducts,andrequire amorecostlyoperationinthesecondperiod.Therefore,theexpectedtotalprotsunder SAP-RCR arehigherthanthoseunder SAP-CR and SAP .FromFigures5-7and5-8, itispossibletoseethatsimilarinterpretationsholdevenwhentheoperationalcostsdo notchangefromperiodtoperiod. 107

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CHAPTER6 CONCLUSIONANDFUTURERESEARCH Inthisdissertation,weinvestigatedtheacquisitionandallocationdecisions,andthe valueofthediagnostictestingandyieldinformationforademanufacturerinhiseffortsto reclaimenvironmentalandeconomicalvaluesresidinginaparticularend-of-lifeproduct duetoareusablepartonitthatcanbesoldintheused-partsmarket.Motivatedbythe currentpracticesaswellasevolvingregulatoryneedsofthedemanufacturingindustryin theUS,weconsidered:ithetrade-offbetweentheenvironmentalfeesandoperational costs,iiacquisitionprice-dependentsupplyofend-of-lifeproducts,iiistochastic demandforreusableparts,ivstochasticdemanufacturingyieldrate,andvrealistic diagnostictestingschemeswithclassicationerrors,thathavenotbeenornotjointly consideredintheexistingliterature. Intherstpartofourstudy,wecharacterizedtheoptimalacquisitionandstocking policyusingathree-stock-pointsetting,whereacquiredend-of-lifeproductscanbe keptinseparatestockpointsfor:end-of-lifeproducts,disassembledbutnotrecovered parts,andrecoveredparts.Weformulatedthreeandsingle-stock-pointmodelswith single-period,prot-maximizingnewsvendormodelswithlostsalesunderfourdifferent settings:acquisition-price-independentordependentsupply,deterministicorstochastic demanufacturingyieldrates.Weshowed,analytically,usingKKTconditions;and numericallybyacomputationalstudy,thatthethree-stock-pointmodeloutperforms anyothersingle-stock-pointmodeleitherintheformofstockingonlytheend-of-life productsorthereusablepartsdisassembledorrecovered.Inaddition,wederivedan explicitlowerboundontheexpecteddemanufacturingyieldrate,whichisnecessary forthedemanufacturertobeabletooperateandmayhelpselectinghissupplysource accordingly.Anextensionofthismodelmayincludecharacterizationoftheacquisition andallocationpolicyconsideringtheproductstructureforaparticularandmorecomplex end-of-lifeproductwithmultipleindependentdemandstreams,i.e.,recoverableparts. 108

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Inaddition,otherdependencieswithrespecttoproductanddisassembly-operation structuresthatdenecost-addingschemesofdisassemblyoperationsanddemand substitutionpossibilitiesmaybeexplored. Inthesecondpartofourstudy,weutilizedtheanalyticalresultsofourrst research,andfocusedontheassessmentofthevalueofthediagnostictesting anddemanufacturingyieldinformationontheprotabilityofthedemanufacturer's business.Tothisextent,weinvestigatedtwodiagnostictestingschemes,whichare subjecttoclassicationerrors,andabenchmarkcasewherenodiagnostictestingis conducted.Thetwodiagnostictestingschemesweconsideredarethepreliminary testingscheme,inwhichthetestisconducteduponreceiptoftheend-of-lifeproducts, andbeforeallocationoccurs;andthepostponedtestingscheme,inwhichthetestis conductedafterthedisassemblyoperationiscompleted.Wenumericallyobserved thattheeconomicalattractivenessofconductingadiagnostictestdependssignicantly ontheleveloftestingerrors,andthenthestagingofthetest,thesupplytypeofthe end-of-lifeproducts,andtoalesserdegreethedemanufacturingyieldrateandtesting cost.Aworthwhileextensionmayincludenon-homogenousdemanufacturingyieldrate distributionforthereusableparts. Inthenalpartofourstudy,wefocusedonatwo-periodextensionofourrst researchproblemtoincludethemarketlifecycleconsiderationoftheend-of-lifeproduct andthereusablepartonit.Westudiedthreepracticallyimplementablemyopicpolicies assumingaminordesignchangeonthereusablepartinthenextsalesperiod,by whichthedemanufacturermaymakeuseofhisexistingstocksofend-of-lifeproducts, andrecoveredpartstosatisfythedemandforthenewgenerationofthepartinthe nextperiod,andhence,maybeabletosavesomeofthecostse.g.,acquisitioncosts, environmentalfees,andoperationalcosts.Wesimpliedouroriginalproblemsetting, andconsideredapreliminarydiagnostictestingschemewithnoclassicationerrors,and twostockpointsinsteadofthree,wheredisassemblyandrecoveryarecombinedinto 109

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asingleoperation,anddeterministicdemanufacturingyieldrate.Themyopicpolicies weconsideredtreattherstperiodasastand-aloneperiodinwhichacquisitionand allocationdecisionsignorethesecondperiod.Therstmyopicpolicyassumesboth periodsareindependentwithnoinventorycarry-overfromtherstperiodintothe next,i.e.,allremainingstockissoldtoamaterialrecycler.Inthesecondperiod,the secondmyopicpolicyassumesthattherecoveredpartsremaininginstockattheend ofperiodonearecarried-over,andtheleft-overstockofend-of-lifeproductsaresoldto thematerialrecycler,whereasthethirdmyopicpolicyassumesthatend-of-lifeproducts remaininginstockattheendoftherstperiodaredemanufactured,andcarried-over tothenextperiodalongwiththeremainingstockofrecoveredparts.Weanalytically characterizedtheformsoftheoptimalwithin-periodsolutionsunderallmyopicpolicies; andshowedthattheoptimalacquisitionandallocationpoliciesineachperiodcan efciently,andeasilybeobtainedbasedonsimpledecisionrulesobtainedfromtheKKT conditions.Theacquisitionquantitiesinbothperiodsarebasedonorder-up-tolevels determinedbytheprobabilitiesofnotstockingout,andalsoontheinitialinventoryin thesecondperiod.Duetouncertaintiesassociatedwiththedemanddistributionsinthe twoperiods,itiscomplicatedtosolvetheoveralltwo-periodmodeloptimally.However, analyticalsolutionsmaybesoughtforunderspecicassumptionsaboutthedemand distributionsinthetwoperiods,whichmaybeafutureextension. 110

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APPENDIXA LOWERBOUNDSONTHEEXPECTEDDEMANUFACTURINGYIELDRATE TableA-1providesthelowerboundsontheexpecteddemanufacturingyieldrate foreachmodelandapplicableKKTcaseseeexamplesinFiguresA-1andA-2. TableA-1.Lowerboundsontheexpecteddemanufacturingyieldrate. ModelandKKTcaseLowerboundModelandKKTcaseLowerbound S3S C1,C4, S1S-P h + a = b )]TJ/F40 6.9738 Tf 6.227 0 Td [(h p )]TJ/F40 6.9738 Tf 6.226 0 Td [(e )]TJ/F40 6.9738 Tf 6.226 0 Td [(c d )]TJ/F40 6.9738 Tf 6.226 0 Td [(c r i + ISS3S C1,C4, ISS1S-P h + c )]TJ/F40 6.9738 Tf 6.227 0 Td [(h p )]TJ/F40 6.9738 Tf 6.226 0 Td [(e )]TJ/F40 6.9738 Tf 6.226 0 Td [(c d )]TJ/F40 6.9738 Tf 6.226 0 Td [(c r i + D3S C1,C4, D1S-PISD3S C1,C4, ISD1S-P S3S C5, S1S-R h + a = b )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r i + ISS3S C5, ISS1S-R h + c )]TJ/F40 6.9738 Tf 6.227 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r i + D3S C5, D1S-RISD3S C5, ISD1S-R S3S C6, S1S-D h + a = b )]TJ/F40 6.9738 Tf 6.227 0 Td [(d )]TJ/F40 6.9738 Tf 6.226 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r i + ISS3S C6, ISS1S-D h + c )]TJ/F40 6.9738 Tf 6.227 0 Td [(d )]TJ/F40 6.9738 Tf 6.226 0 Td [(c d )]TJ/F40 6.9738 Tf 6.227 0 Td [(c r i + D3S C6, D1S-DISD3S C6, ISD1S-D FigureA-1.Lowerboundsontheexpecteddemanufacturingyieldrateformodels: S3S C1,C4, S1S-P D3S C1,C4, D1S-P e d :H,H, c :L, b :L. FigureA-2.Lowerboundsontheexpecteddemanufacturingyieldrateformodels: ISS3S C1,C4, ISS1S-P ISD3S C1,C4, ISD1S-P e d :H,H, c :L, b : L. 111

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APPENDIXB DERIVATIONSFORSINGLE-STOCK-POINTMODELSUNDERDETERMINISTIC DEMANUFACTURINGYIELDRATE Expectedprotfunctionsforthesingle-stock-pointmodelsunderdeterministic demanufacturingyieldratearegivenbelow: D1S-P : D 1 S )]TJ/F40 7.9701 Tf 6.586 0 Td [(P P = Z P 0 p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r xf x dx + Z 1 P p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.956 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r Pf x dx + Z P 0 v P )]TJ/F39 11.9552 Tf 11.955 0 Td [(x f x dx )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z 1 P r x )]TJ/F39 11.9552 Tf 11.955 0 Td [(P f x dx )]TJ/F22 11.9552 Tf 11.955 0 Td [( P = y P = y )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b )]TJ/F39 11.9552 Tf 11.956 0 Td [(c i P = y )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p P + v h + v )]TJ/F39 11.9552 Tf 11.955 0 Td [(y P = y D1S-D : D 1 S )]TJ/F40 7.9701 Tf 6.587 0 Td [(D D = Z D 0 p )]TJ/F39 11.9552 Tf 11.956 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r xf x dx + Z 1 D p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r Df x dx + Z D 0 v D )]TJ/F39 11.9552 Tf 11.955 0 Td [(x f x dx )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z 1 D r x )]TJ/F39 11.9552 Tf 11.955 0 Td [(D f x dx )]TJ/F22 11.9552 Tf 11.955 0 Td [( D = y D = y )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i D = y )]TJ/F39 11.9552 Tf 11.956 0 Td [(c d D + v h + v )]TJ/F39 11.9552 Tf 11.955 0 Td [(y D = y D1S-R : D 1 S )]TJ/F40 7.9701 Tf 6.586 0 Td [(R R = Z R 0 pxf x dx + Z 1 R pRf x dx + Z R 0 v R )]TJ/F39 11.9552 Tf 11.955 0 Td [(x f x dx )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z 1 R r x )]TJ/F39 11.9552 Tf 11.955 0 Td [(R f x dx )]TJ/F22 11.9552 Tf 11.955 0 Td [( R = y R = y )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i R = y )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d R )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r R + v h + v )]TJ/F39 11.9552 Tf 11.955 0 Td [(y R = y Concavityconditionsobtainedfromthesecondorderconditionsarethesameas thoseinthestochasticdemanufacturingyieldratecase.Firstorderconditionstosolve forthevaluesofthedecisionvariablesareasfollows:i )]TJ/F39 11.9552 Tf 12.782 0 Td [(e )]TJ/F39 11.9552 Tf 12.782 0 Td [(c d )]TJ/F39 11.9552 Tf 12.782 0 Td [(c r yF P + 2 P = by = )]TJ/F39 11.9552 Tf 12.457 0 Td [(h p )]TJ/F39 11.9552 Tf 12.456 0 Td [(e )]TJ/F39 11.9552 Tf 12.457 0 Td [(c d )]TJ/F39 11.9552 Tf 12.457 0 Td [(c r y )]TJ/F22 11.9552 Tf 12.457 0 Td [( )]TJ/F39 11.9552 Tf 12.457 0 Td [(a = b ,ii )]TJ/F39 11.9552 Tf 12.456 0 Td [(d )]TJ/F39 11.9552 Tf 12.456 0 Td [(c r yF D +2 D = by = )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r y )]TJ/F22 11.9552 Tf 11.956 0 Td [( )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b ,andiii yF R +2 R = by = )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r y )]TJ/F22 11.9552 Tf 11.955 0 Td [( )]TJ/F39 11.9552 Tf 11.955 0 Td [(a = b Notethat,whenthesupplyquantityofend-of-lifeproductsdoesnotdependonthe acquisitionprice,itissufcienttoreplace P = y )]TJ/F39 11.9552 Tf 9.978 0 Td [(a = b D = y )]TJ/F39 11.9552 Tf 9.978 0 Td [(a = b ,and R = y )]TJ/F39 11.9552 Tf 9.978 0 Td [(a = b withtheparameter c intheexpectedprotequationsabovetoobtaintheexpectedprot functionsformodels ISD1S-P ISD1S-D ,and ISD1S-R respectively. 112

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APPENDIXC PROBABILITIESOFNOTSTOCKINGOUTFORSINGLEANDTHREE-STOCKPOINT MODELSUNDERUNIFORMDEMAND TableC-1.Probabilitiesofnotstockingout = p )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r = c i )]TJ/F39 11.9552 Tf 11.955 0 Td [(v )]TJ/F39 11.9552 Tf 11.955 0 Td [(v h X Uniform, U Modeland KKTCase F R F R + D F R + D + P S3S C1 dE [ Y ] d + c r E [ Y 2 ] e )]TJ/F40 7.9701 Tf 6.587 0 Td [(d + h p E [ Y ] e )]TJ/F40 7.9701 Tf 6.587 0 Td [(d + c d E [ Y 2 ] )]TJ/F40 7.9701 Tf 6.586 0 Td [(h p )]TJ/F40 7.9701 Tf 6.587 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r E [ Y ] )]TJ/F23 7.9701 Tf 6.586 0 Td [( )]TJ/F40 7.9701 Tf 6.586 0 Td [(a = b )]TJ/F40 7.9701 Tf 6.586 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r E [ Y 2 ]+2 U = b S3S C4 e + h p E [ Y ] e + c d + c r E [ Y 2 ] )]TJ/F40 7.9701 Tf 6.586 0 Td [(h p )]TJ/F40 7.9701 Tf 6.587 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r E [ Y ] )]TJ/F23 7.9701 Tf 6.586 0 Td [( )]TJ/F40 7.9701 Tf 6.586 0 Td [(a = b )]TJ/F40 7.9701 Tf 6.586 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r E [ Y 2 ]+2 U = b S3S C5 )]TJ/F40 7.9701 Tf 6.586 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r E [ Y ] )]TJ/F23 7.9701 Tf 6.587 0 Td [( )]TJ/F40 7.9701 Tf 6.586 0 Td [(a = b E [ Y 2 ]+2 U = b S3S C6 dE [ Y ] d + c r E [ Y 2 ] )]TJ/F40 7.9701 Tf 6.586 0 Td [(d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r r E [ Y ] )]TJ/F23 7.9701 Tf 6.587 0 Td [( )]TJ/F40 7.9701 Tf 6.586 0 Td [(a = b )]TJ/F40 7.9701 Tf 6.586 0 Td [(d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r E [ Y 2 ]+2 U = b S1S-P )]TJ/F40 7.9701 Tf 6.586 0 Td [(h p )]TJ/F40 7.9701 Tf 6.587 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r r E [ Y ] )]TJ/F23 7.9701 Tf 6.587 0 Td [( )]TJ/F40 7.9701 Tf 6.586 0 Td [(a = b )]TJ/F40 7.9701 Tf 6.587 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r E [ Y 2 ]+2 U = b S1S-D )]TJ/F40 7.9701 Tf 6.586 0 Td [(d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r E [ Y ] )]TJ/F23 7.9701 Tf 6.586 0 Td [( )]TJ/F40 7.9701 Tf 6.586 0 Td [(a = b )]TJ/F40 7.9701 Tf 6.586 0 Td [(d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r E [ Y 2 ]+2 U = b S1S-R )]TJ/F40 7.9701 Tf 6.586 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r E [ Y ] )]TJ/F23 7.9701 Tf 6.587 0 Td [( )]TJ/F40 7.9701 Tf 6.586 0 Td [(a = b E [ Y 2 ]+2 U = b D3S C1 d d + c r e )]TJ/F40 7.9701 Tf 6.586 0 Td [(d + h p e )]TJ/F40 7.9701 Tf 6.587 0 Td [(d + c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(h p )]TJ/F40 7.9701 Tf 6.586 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r )]TJ/F23 7.9701 Tf 6.587 0 Td [( )]TJ/F40 7.9701 Tf 6.587 0 Td [(a = b = y )]TJ/F40 7.9701 Tf 6.586 0 Td [(e )]TJ/F40 7.9701 Tf 6.586 0 Td [(c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r +2 U = by 2 D3S C4 e + h p e + c d + c r )]TJ/F40 7.9701 Tf 6.586 0 Td [(h p )]TJ/F40 7.9701 Tf 6.587 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r )]TJ/F23 7.9701 Tf 6.586 0 Td [( )]TJ/F40 7.9701 Tf 6.587 0 Td [(a = b = y )]TJ/F40 7.9701 Tf 6.586 0 Td [(e )]TJ/F40 7.9701 Tf 6.586 0 Td [(c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r +2 U = by 2 D3S C5 )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r )]TJ/F23 7.9701 Tf 6.586 0 Td [( )]TJ/F40 7.9701 Tf 6.586 0 Td [(a = b = y +2 U = by 2 D3S C6 d d + c r )]TJ/F40 7.9701 Tf 6.586 0 Td [(d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r )]TJ/F23 7.9701 Tf 6.586 0 Td [( )]TJ/F40 7.9701 Tf 6.587 0 Td [(a = b = y )]TJ/F40 7.9701 Tf 6.587 0 Td [(d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r +2 U = by 2 D1S-P )]TJ/F40 7.9701 Tf 6.586 0 Td [(h p )]TJ/F40 7.9701 Tf 6.586 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r )]TJ/F23 7.9701 Tf 6.587 0 Td [( )]TJ/F40 7.9701 Tf 6.587 0 Td [(a = b = y )]TJ/F40 7.9701 Tf 6.586 0 Td [(e )]TJ/F40 7.9701 Tf 6.586 0 Td [(c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r +2 U = by 2 D1S-D )]TJ/F40 7.9701 Tf 6.586 0 Td [(d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r )]TJ/F23 7.9701 Tf 6.586 0 Td [( )]TJ/F40 7.9701 Tf 6.587 0 Td [(a = b = y )]TJ/F40 7.9701 Tf 6.587 0 Td [(d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r +2 U = by 2 D1S-R )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r )]TJ/F23 7.9701 Tf 6.586 0 Td [( )]TJ/F40 7.9701 Tf 6.586 0 Td [(a = b = y +2 U = by 2 ISS3S C1 dE [ Y ] d + c r E [ Y 2 ] e )]TJ/F40 7.9701 Tf 6.587 0 Td [(d + h p E [ Y ] e )]TJ/F40 7.9701 Tf 6.587 0 Td [(d + c d E [ Y 2 ] )]TJ/F40 7.9701 Tf 6.586 0 Td [(h p )]TJ/F40 7.9701 Tf 6.587 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r E [ Y ] )]TJ/F23 7.9701 Tf 6.586 0 Td [( + c )]TJ/F40 7.9701 Tf 6.587 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r E [ Y 2 ] ISS3S C4 e + h p E [ Y ] e + c d + c r E [ Y 2 ] )]TJ/F40 7.9701 Tf 6.586 0 Td [(h p )]TJ/F40 7.9701 Tf 6.587 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r E [ Y ] )]TJ/F23 7.9701 Tf 6.586 0 Td [( + c )]TJ/F40 7.9701 Tf 6.587 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r E [ Y 2 ] ISS3S C5 )]TJ/F40 7.9701 Tf 6.586 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r E [ Y ] )]TJ/F23 7.9701 Tf 6.587 0 Td [( + c ISS3S C6 dE [ Y ] d + c r E [ Y 2 ] )]TJ/F40 7.9701 Tf 6.586 0 Td [(d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r E [ Y ] )]TJ/F23 7.9701 Tf 6.586 0 Td [( + c )]TJ/F40 7.9701 Tf 6.587 0 Td [(d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r ISS1S-P )]TJ/F40 7.9701 Tf 6.586 0 Td [(h p )]TJ/F40 7.9701 Tf 6.587 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r E [ Y ] )]TJ/F23 7.9701 Tf 6.586 0 Td [( + c )]TJ/F40 7.9701 Tf 6.587 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r E [ Y 2 ] ISS1S-D )]TJ/F40 7.9701 Tf 6.586 0 Td [(d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r E [ Y ] )]TJ/F23 7.9701 Tf 6.586 0 Td [( + c )]TJ/F40 7.9701 Tf 6.586 0 Td [(d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r E [ Y 2 ] ISS1S-R )]TJ/F40 7.9701 Tf 6.586 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r E [ Y ] )]TJ/F23 7.9701 Tf 6.587 0 Td [( + c E [ Y 2 ] ISD3S C1 d d + c r e )]TJ/F40 7.9701 Tf 6.586 0 Td [(d + h p e )]TJ/F40 7.9701 Tf 6.587 0 Td [(d + c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(h p )]TJ/F40 7.9701 Tf 6.586 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r )]TJ/F23 7.9701 Tf 6.587 0 Td [( + c = y )]TJ/F40 7.9701 Tf 6.587 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r ISD3S C4 e + h p e + c d + c r )]TJ/F40 7.9701 Tf 6.586 0 Td [(h p )]TJ/F40 7.9701 Tf 6.587 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r )]TJ/F23 7.9701 Tf 6.586 0 Td [( + c = y )]TJ/F40 7.9701 Tf 6.587 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r ISD3S C5 )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r )]TJ/F23 7.9701 Tf 6.586 0 Td [( + c = y ISD3S C6 d d + c r )]TJ/F40 7.9701 Tf 6.586 0 Td [(d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r )]TJ/F23 7.9701 Tf 6.586 0 Td [( + c = y )]TJ/F40 7.9701 Tf 6.586 0 Td [(d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r ISD1S-P )]TJ/F40 7.9701 Tf 6.586 0 Td [(h p )]TJ/F40 7.9701 Tf 6.587 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r )]TJ/F23 7.9701 Tf 6.586 0 Td [( + c = y )]TJ/F40 7.9701 Tf 6.587 0 Td [(e )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r ISD1S-D )]TJ/F40 7.9701 Tf 6.586 0 Td [(d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r )]TJ/F23 7.9701 Tf 6.586 0 Td [( + c = y )]TJ/F40 7.9701 Tf 6.586 0 Td [(d )]TJ/F40 7.9701 Tf 6.586 0 Td [(c r ISD1S-R )]TJ/F40 7.9701 Tf 6.586 0 Td [(c d )]TJ/F40 7.9701 Tf 6.587 0 Td [(c r )]TJ/F23 7.9701 Tf 6.586 0 Td [( + c = y 113

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APPENDIXD LOWERBOUNDSONTHEEXPECTEDDEMANUFACTURINGYIELDRATEFOR DIAGNOSTICTESTINGANDNOTESTINGMODELS TableD-1providesthelowerboundsontheexpecteddemanufacturingyieldrate foreachmodelandapplicableKKTcaseseeexamplesinFiguresD-1andD-2. TableD-1.Lowerboundsontheexpecteddemanufacturingyieldratefordiagnostic testingandnotestingmodels. ModelandKKTcaseLowerboundModelandKKTcaseLowerbound PTS C1,C4 h c i + + a = b + )]TJ/F10 6.9738 Tf 6.227 0 Td [( c d + c r + h p )]TJ/F40 6.9738 Tf 6.226 0 Td [(e + )]TJ/F10 6.9738 Tf 6.227 0 Td [( )]TJ/F10 6.9738 Tf 6.227 0 Td [( c d + c r + h p i + ISPTS C1,C4 h c i + + c + )]TJ/F10 6.9738 Tf 6.227 0 Td [( c d + c r + h p )]TJ/F40 6.9738 Tf 6.227 0 Td [(e + )]TJ/F10 6.9738 Tf 6.227 0 Td [( )]TJ/F10 6.9738 Tf 6.227 0 Td [( c d + c r + h p i + PTD C1,C4 ISPTD C1,C4 PTS C5 h c i + + a = b + )]TJ/F10 6.9738 Tf 6.227 0 Td [( c d + c r + )]TJ/F10 6.9738 Tf 6.226 0 Td [( )]TJ/F10 6.9738 Tf 6.226 0 Td [( c d + c r i + ISPTS C5 h c i + + c + )]TJ/F10 6.9738 Tf 6.226 0 Td [( c d + c r + )]TJ/F10 6.9738 Tf 6.226 0 Td [( )]TJ/F10 6.9738 Tf 6.226 0 Td [( c d + c r i + PTD C5 ISPTD C5 PTS C6 h c i + + a = b + )]TJ/F10 6.9738 Tf 6.227 0 Td [( c d + c r )]TJ/F40 6.9738 Tf 6.226 0 Td [(d + )]TJ/F10 6.9738 Tf 6.226 0 Td [( )]TJ/F10 6.9738 Tf 6.226 0 Td [( c d + c r i + ISPTS C6 h c i + + c + )]TJ/F10 6.9738 Tf 6.227 0 Td [( c d + c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(d + )]TJ/F10 6.9738 Tf 6.227 0 Td [( )]TJ/F10 6.9738 Tf 6.227 0 Td [( c d + c r i + PTD C6 ISPTD C6 DTS C1,C4 h c i + + a = b + h p + c d + )]TJ/F10 6.9738 Tf 6.226 0 Td [( c r )]TJ/F40 6.9738 Tf 6.226 0 Td [(e + )]TJ/F10 6.9738 Tf 6.226 0 Td [( )]TJ/F10 6.9738 Tf 6.226 0 Td [( c r i + ISDTS C1,C4 h c i + + c + h p + c d + )]TJ/F10 6.9738 Tf 6.227 0 Td [( c r )]TJ/F40 6.9738 Tf 6.226 0 Td [(e + )]TJ/F10 6.9738 Tf 6.226 0 Td [( )]TJ/F10 6.9738 Tf 6.226 0 Td [( c r i + DTD C1,C4 ISDTD C1,C4 DTS C5 h c i + + a = b + c d + )]TJ/F10 6.9738 Tf 6.227 0 Td [( c r + )]TJ/F10 6.9738 Tf 6.227 0 Td [( )]TJ/F10 6.9738 Tf 6.227 0 Td [( c r i + ISDTS C5 h c i + + c + c d + )]TJ/F10 6.9738 Tf 6.227 0 Td [( c r + )]TJ/F10 6.9738 Tf 6.227 0 Td [( )]TJ/F10 6.9738 Tf 6.227 0 Td [( c r i + DTD C5 ISDTD C5 DTS C6 h c i + + a = b + c d + )]TJ/F10 6.9738 Tf 6.227 0 Td [( c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(d + )]TJ/F10 6.9738 Tf 6.227 0 Td [( )]TJ/F10 6.9738 Tf 6.227 0 Td [( c r i + ISDTS C6 h c i + + c + c d + )]TJ/F10 6.9738 Tf 6.227 0 Td [( c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(d + )]TJ/F10 6.9738 Tf 6.227 0 Td [( )]TJ/F10 6.9738 Tf 6.227 0 Td [( c r i + DTD C6 ISDTD C6 NTS C1,C4 h + a = b + c d + c r + h p )]TJ/F40 6.9738 Tf 6.227 0 Td [(e i + ISNTS C1,C4 h + c + c d + c r + h p )]TJ/F40 6.9738 Tf 6.227 0 Td [(e i + NTD C1,C4 ISNTD C1,C4 NTS C5 h + a = b + c d + c r i + ISNTS C5 h + c + c d + c r i + NTD C5 ISNTD C5 NTS C6 h + a = b + c d + c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(d i + ISNTS C6 h + c + c d + c r )]TJ/F40 6.9738 Tf 6.227 0 Td [(d i + NTD C6 ISNTD C6 FigureD-1.Lowerboundsontheexpecteddemanufacturingyieldrateformodels: PTS and PTD C1,C4, e d :H,H, c :L, b :L, =.85,0.80 114

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FigureD-2.Lowerboundsontheexpecteddemanufacturingyieldrateformodels: ISPTS and ISPTD C1,C4, e d :H,H, c :L, b :L, =.85,0.80 115

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APPENDIXE KKTSOLUTIONSFORACQUISITION-PRICE-INDEPENDENTSUPPLYMODELS UNDERSTOCHASTICDEMANUFACTURINGYIELDRATE FirstorderconditionsareusedtondthesolutionsunderapplicableKKTcases. Theseconditionsaregivenbelowfor ISPTS ISDTS ,and ISNTS : ISPTS 0 R = )]TJ/F30 11.9552 Tf 11.291 16.272 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [( y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( c d + c r F y R + D + P g y dy )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.48 -9.683 Td [( y e )]TJ/F39 11.9552 Tf 11.955 0 Td [(d + c d F y R + D g y dy )]TJ/F30 11.9552 Tf 11.955 16.272 Td [(Z y U y L yd + c r F yR g y dy + E [ Y ] p )]TJ/F39 11.9552 Tf 11.956 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c p )]TJ/F22 11.9552 Tf 11.955 0 Td [( c d + c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h ISPTS 0 D = )]TJ/F30 11.9552 Tf 11.291 16.272 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( y p )]TJ/F39 11.9552 Tf 11.956 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( c d + c r F y R + D + P g y dy )]TJ/F30 11.9552 Tf 11.956 16.273 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( y e )]TJ/F39 11.9552 Tf 11.955 0 Td [(d + c d F y R + D g y dy + E [ Y ] p )]TJ/F39 11.9552 Tf 11.955 0 Td [(d )]TJ/F39 11.9552 Tf 11.956 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c p )]TJ/F22 11.9552 Tf 11.955 0 Td [( c d + c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h ISPTS 0 P = )]TJ/F30 11.9552 Tf 11.291 16.273 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( y p )]TJ/F39 11.9552 Tf 11.956 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( c d + c r F y R + D + P g y dy + E [ Y ] p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c p )]TJ/F22 11.9552 Tf 11.955 0 Td [( c d + c r + h p )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h ISDTS 0 R = )]TJ/F30 11.9552 Tf 11.291 16.272 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i )]TJ/F25 11.9552 Tf 11.955 0 Td [( c r F y R + D + P g y dy )]TJ/F30 11.9552 Tf 11.956 16.272 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( y e )]TJ/F39 11.9552 Tf 11.955 0 Td [(d + c d + c i F y R + D g y dy )]TJ/F30 11.9552 Tf 11.956 16.273 Td [(Z y U y L yd + c r F yR g y dy + E [ Y ] p )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c p )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F22 11.9552 Tf 11.955 0 Td [( c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h ISDTS 0 D = )]TJ/F30 11.9552 Tf 11.291 16.273 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [( y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i )]TJ/F25 11.9552 Tf 11.955 0 Td [( c r F y R + D + P g y dy )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [( y e )]TJ/F39 11.9552 Tf 11.956 0 Td [(d + c d + c i F y R + D g y dy 116

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+ E [ Y ] p )]TJ/F39 11.9552 Tf 11.956 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c p )]TJ/F39 11.9552 Tf 11.956 0 Td [(c d )]TJ/F22 11.9552 Tf 11.955 0 Td [( c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h ISDTS 0 P = )]TJ/F30 11.9552 Tf 11.291 16.272 Td [(Z y U y L )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [( y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.956 0 Td [(v + r )]TJ/F25 11.9552 Tf 11.955 0 Td [( c d + c r F y R + D + P g y dy + E [ Y ] p )]TJ/F39 11.9552 Tf 11.956 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p )]TJ/F39 11.9552 Tf 11.955 0 Td [(c p )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F22 11.9552 Tf 11.955 0 Td [( c r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c i + v + v h ISNTS 0 R = )]TJ/F30 11.9552 Tf 11.291 16.273 Td [(Z y U y L )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.956 0 Td [(c r F y R + D + P g y dy )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z y U y L )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(y e )]TJ/F39 11.9552 Tf 11.955 0 Td [(d + c d F y R + D g y dy )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z y U y L yd + c r F yR g y dy )]TJ/F39 11.9552 Tf 11.955 0 Td [(c p + E [ Y ] p )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r + v + v h ISNTS 0 D = )]TJ/F30 11.9552 Tf 11.291 16.272 Td [(Z y U y L )]TJ/F39 11.9552 Tf 5.479 -9.683 Td [(y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r F y R + D + P g y dy )]TJ/F30 11.9552 Tf 11.956 16.272 Td [(Z y U y L )]TJ/F39 11.9552 Tf 5.48 -9.684 Td [(y e )]TJ/F39 11.9552 Tf 11.956 0 Td [(d + c d F y R + D g y dy )]TJ/F39 11.9552 Tf 11.956 0 Td [(c p + E [ Y ] p )]TJ/F39 11.9552 Tf 11.956 0 Td [(d )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r + v + v h ISNTS 0 P = )]TJ/F30 11.9552 Tf 11.291 16.272 Td [(Z y U y L )]TJ/F39 11.9552 Tf 5.479 -9.684 Td [(y p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.956 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r F y R + D + P g y dy )]TJ/F39 11.9552 Tf 11.955 0 Td [(c p )]TJ/F39 11.9552 Tf 11.955 0 Td [(h p + E [ Y ] p )]TJ/F39 11.9552 Tf 11.955 0 Td [(e )]TJ/F39 11.9552 Tf 11.955 0 Td [(v + r )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(c r + v + v h ThesolutionstotheKKTcasesareobtainedsolvingasystemofanappropriate setofrstorderconditionsundereachcase,andaregiveninTablesE-1,E-2,andE-3 underthegeneraldemanddistribution: TableE-1.ISPTS:KKTcasesunderstochasticdemanufacturingyieldrateandgeneral demanddistribution. CaseEquationstosolve C1 E [ Yd + c r F YR ]= E [ Y ] d E [ Y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d F Y R + D ]= E [ Y ] e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.963 0 Td [( c d + c r F Y R + D + P ] = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.962 0 Td [( c d + c r + h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h C4 E [ Ye + c d + c r F YR ]= E [ Y ] e + h p E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.963 0 Td [( c d + c r F Y R + P ] = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.962 0 Td [( c d + c r + h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h C5 E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r F YR ]= E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.962 0 Td [( c d + c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h C6 E [ Yd + c r F YR ]= E [ Y ] d E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F11 9.9626 Tf 9.962 0 Td [( c r F Y R + D ]= E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.963 0 Td [( c d + c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c p )]TJ/F39 9.9626 Tf 7.749 0 Td [(c i + v + v h 117

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TableE-2.ISDTS:KKTcasesunderstochasticdemanufacturingyieldrateandgeneral demanddistribution. CaseEquationstosolve C1 E [ Yd + c r F YR ]= E [ Y ] d E [ Y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d + c i F Y R + D ]= E [ Y ] e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i )]TJ/F11 9.9626 Tf 9.963 0 Td [( c r F Y R + D + P ] = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F22 9.9626 Tf 9.962 0 Td [( c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h C4 E [ Ye + c d + c i + c r F YR ]= E [ Y ] e + h p E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i )]TJ/F22 9.9626 Tf 9.963 0 Td [( c r F Y R + P ] = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F22 9.9626 Tf 9.962 0 Td [( c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h C5 E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r F YR ]= E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F22 9.9626 Tf 9.962 0 Td [( c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h C6 E [ Yd + c r F YR ]= E [ Y ] d E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.962 0 Td [( c r F Y R + D ] = E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F22 9.9626 Tf 9.963 0 Td [( c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h TableE-3.ISNTS:KKTcasesunderstochasticdemanufacturingyieldrateandgeneral demanddistribution. CaseEquationstosolve C1 E [ Yd + c r F YR ]= E [ Y ] d E [ Y e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + c d F Y R + D ]= E [ Y ] e )]TJ/F39 9.9626 Tf 9.962 0 Td [(d + h p E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r F Y R + D + P ] = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c r + v + v h C4 E [ Ye + c d + c r F YR ]= E [ Y ] e + h p E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c r F Y R + P ] = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c r + v + v h C5 E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r F YR ]= E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c d )]TJ/F39 9.9626 Tf 9.962 0 Td [(c r + v + v h C6 E [ Yd + c r F YR ]= E [ Y ] d E [ Y p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c r F Y R + D ]= E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r + v + v h Explicitsolutionstotherstorderconditionscanbeobtainedwhenthedemand distributionisuniform.SolutionstotheKKTcasesforthesemodelsaregivenin TablesE-4,E-5,andE-6undertheuniformlydistributeddemand: TableE-4.ISPTS:KKTcasesunderstochasticdemanufacturingyieldrateanduniform U demand. CaseEquationstosolve C1 F R = E [ Y ] d = E [ Y 2 ] d + E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.963 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.962 0 Td [(E [ Y 2 ] c r F R + D = E [ Y ] e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p = E [ Y 2 ] e )]TJ/F39 9.9626 Tf 9.962 0 Td [(d + E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.963 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.962 0 Td [(E [ Y 2 ] c d F R + D + P ]= E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.963 0 Td [( c d + c r + h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.963 0 Td [( E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.963 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.962 0 Td [(E [ Y 2 ] c d + c r C4 F R = E [ Y ] e + h p = E [ Y 2 ] e + E [ Y 2 ]+1 )]TJ/F11 9.9626 Tf 9.962 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y 2 ] c d + c r F R + P = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.963 0 Td [( c d + c r + h p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.963 0 Td [( E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.963 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.962 0 Td [(E [ Y 2 ] c d + c r C5 F R = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.963 0 Td [( c d + c r )]TJ/F39 9.9626 Tf 9.962 0 Td [(c p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h = 2 E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r C6 F R = E [ Y ] d = E [ Y 2 ] d + E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.963 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.962 0 Td [(E [ Y 2 ] c r F R + D = E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.963 0 Td [( c d + c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c i + v + v h = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.962 0 Td [( E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.962 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y 2 ] c r 118

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TableE-5.ISDTS:KKTcasesunderstochasticdemanufacturingyieldrateanduniform U demand. CaseEquationstosolve C1 F R = E [ Y ] d = E [ Y 2 ] d + E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.963 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.962 0 Td [(E [ Y 2 ] c r F R + D = E [ Y ] e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p = E [ Y 2 ] e )]TJ/F39 9.9626 Tf 9.962 0 Td [(d + E [ Y ] c d + c i F R + D + P ]= E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F22 9.9626 Tf 9.963 0 Td [( c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.962 0 Td [( E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.962 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y 2 ] c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y ] c d + c i C4 F R = E [ Y ] e + h p = E [ Y 2 ] e + E [ Y 2 ]+1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y 2 ] c r + E [ Y ] c d + c i F R + P = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F22 9.9626 Tf 9.963 0 Td [( c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.962 0 Td [( E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.962 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y 2 ] c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y ] c d + c i C5 F R = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F22 9.9626 Tf 9.963 0 Td [( c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h = 2 E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r C6 F R = E [ Y ] d = E [ Y 2 ] d + E [ Y 2 ]+ )]TJ/F11 9.9626 Tf 9.963 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.962 0 Td [(E [ Y 2 ] c r F R + D = E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F22 9.9626 Tf 9.963 0 Td [( c r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c i + v + v h = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F22 9.9626 Tf 9.962 0 Td [( E [ Y 2 ]+1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( E [ Y ] )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y 2 ] c r TableE-6.ISNTS:KKTcasesunderstochasticdemanufacturingyieldrateanduniform U demand. CaseEquationstosolve C1 F R = E [ Y ] d = E [ Y 2 ] d + E [ Y ] c r F R + D = E [ Y ] e )]TJ/F39 9.9626 Tf 9.963 0 Td [(d + h p = E [ Y 2 ] e )]TJ/F39 9.9626 Tf 9.962 0 Td [(d + E [ Y ] c d F R + D + P ]= E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r + v + v h = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y ] c d + c r C4 F R = E [ Y ] e + h p = E [ Y 2 ] e + E [ Y ] c d + c r F R + P = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(h p )]TJ/F39 9.9626 Tf 9.963 0 Td [(c p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r + v + v h = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(e )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(E [ Y ] c d + c r C5 F R = E [ Y ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r + v + v h = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r C6 F R = E [ Y ] d = E [ Y 2 ] d + E [ Y ] c r F R + D = E [ Y ] p )]TJ/F39 9.9626 Tf 9.962 0 Td [(d )]TJ/F39 9.9626 Tf 9.963 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.963 0 Td [(c p )]TJ/F39 9.9626 Tf 9.962 0 Td [(c d )]TJ/F39 9.9626 Tf 9.963 0 Td [(c r + v + v h = E [ Y 2 ] p )]TJ/F39 9.9626 Tf 9.963 0 Td [(d )]TJ/F39 9.9626 Tf 9.962 0 Td [(v + r )]TJ/F39 9.9626 Tf 9.962 0 Td [(E [ Y ] c r 119

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APPENDIXF TOTALACQUISITIONQUANTITYANDPRICEUNDERTESTINGANDNOTESTING SCENARIOS A B C D E F G H I FigureF-1.Comparisonoftotalacquisitionquantitiesunderacquisition-price-dependent supply,stochasticyieldmodels, a b =%,30% c i = L c d + c r = L A,D,G = =0.85 .B,E,H = =0.90 .C,F,I = =1 120

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A B C D E F G H I FigureF-2.Comparisonoftotalacquisitionquantitiesunder acquisition-price-independentsupply,stochasticyieldmodels, a b =%,30% c i = L c d + c r = L .A,D,G = =0.85 B,E,H = =0.90 .C,F,I = =1 121

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A B C D E F G H I FigureF-3.Comparisonoftotalacquisitionpricesunderacquisition-price-dependent supply,stochasticyieldmodels, a b =%,30% c i = L c d + c r = L A,D,G = =0.85 .B,E,H = =0.90 .C,F,I = =1 122

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Karakayali,I.,H.Emir-Farinas,E.Akcali.2007.Ananalysisofdecentralizedcollection andprocessingofend-of-lifeproducts. JournalofOperationsManagement 25 1161. Karakayal, I.,H.Emir-Farinas,E.Akcal.2010.Pricingandrecoveryplanningfor demanufacturingoperationswithmultipleusedproductsandmultiplereusable components. Computers&IndustrialEngineering 59 55. Ketzenberg,M.E.,E.vanderLaan,R.H.Teunter.2004.Thevalueofinformationin reverselogistics.Tech.Rep.ERS-2004-053-LIS,ErasmusResearchInstituteof Management,RotterdamSchoolofManagement/RotterdamSchoolofEconomics. Khouja,M.1999.Thesingle-periodnews-vendorproblem:literaturereviewand suggestionsforfutureresearch. Omega,InternationalJournalofManagement Science 27 537. Krikke,H.R.,A.vanHarten,P.C.Schuur.1998.Onamediumtermproductrecovery anddisposalstrategyfordurableassemblyproducts. InternationalJournalof ProductionResearch 36 111. Langella,I.M.2007.Heuristicsfordemand-drivendisassemblyplanning. Computers& OperationsResearch 34 552. Ozdemir, O.,M.Denizel,V.D.R.Guide,Jr.2011.Recoverydecisionsofaproducerin alegislativedisposalfeeenvironment. EuropeanJournalofOperationalResearch 216 293. Palmer,K.,M.Walls.1999.Extendedproductresponsibility:Aneconomicassessment ofalternativepolicies.DiscussionPapersdp-99-12,ResourcesFortheFuture. Regulations.2011.Retrievedfromhttp://www.epa.gov/lawsregs/regulations/index.html. Taleb,K.N.,S.M.Gupta.1997.Disassemblyofmultipleproductstructures. Computers &IndustrialEngineering 32 949. Teunter,R.H.2006.Determiningoptimaldisassemblyandrecoverystrategies. Omega 34 533. Teunter,R.H.,S.D.P.Flapper.2011.Optimalcoreacquisitionandremanufacturing policiesunderuncertaincorequalityfractions. EuropeanJournalofOperational Research 210 241. vanderLaan,E.,G.P.Kiesm uller,R.Kuik,D.Vlachos,R.Dekker.2004.Stochastic inventorycontrolforproductrecovery.R.Dekker,M.Fleischmann,K.Inderfurth, L.VanWassenhove,eds., ReverseLogistics:QuantitativeModelsforClosed-Loop SupplyChains .Springer,181. Veinott,A.,Jr.1965a.Optimalpolicyforamulti-product,dynamic,nonstationary inventoryproblem. ManagementScience 12 206. 125

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BIOGRAPHICALSKETCH GoncaYldrmearnedherB.S.degreein1998,andM.S.degreein2001,in industrialengineeringfromBilkentUniversityinAnkara,Turkey.Sheworkedasa full-timeinstructorintheDepartmentofIndustrialEngineeringatCankayaUniversity, Ankara,Turkey,duringtheacademicyearsfrom1999until2004,whereshetaught undergraduatelevelclassesincludingintroductoryoperationsresearch,probabilityand statistics,andsimulation.ShebeganheracademicstudyintheDepartmentofIndustrial andSystemsEngineeringattheUniversityofFloridainAugust2006. 127